UFL/COEL/MP94/01
FLUID MUD AND WATER WAVES: A BRIEF REVIEW
OF INTERACTIVE PROCESSES AND SIMPLE
MODELING APPROACHES
by
Ashish J. Mehta
SayChong Lee
and
Yigong Li
February, 1994
UFL/COEL/MP94/01
FLUID MUD AND WATER WAVES: A BRIEF REVIEW OF INTERACTIVE PROCESSES
AND SIMPLE MODELING APPROACHES
Ashish J. Mehta
SayChong Lee
and
Yigong Li
Coastal and Oceanographic Engineering Department
University of Florida, Gainesville, FL 32611
February, 1994
TABLE OF CONTENTS
LIST OF FIGURES ...................... .................................. 3
LIST OF TABLES ...................................... .................. 7
SYNOPSIS ................... .......... ................................... 8
INTRODUCTION ...................................... ................... 9
1.1 MUDWAVE DYNAMICS IN ENGINEERING AND SCIENTIFIC APPLICATIONS ....... 9
1.2 PROCESSES AND FEEDBACKS ................. .......... ........ ..... 9
MUDW AVE DOMAIN ...................... ........ .........................
2.1 DEFINITIONS ..... .............................................
2.2 FLUID M UD .................. .. .................................
GENERATION AND EROSION OF FLUID MUD .....................................
3.1 FLUIDIZATION ................................ .................
3.2 ESTIMATION OF FLUIDIZATION THICKNESS .... .........................
3.3 SURFACE EROSION ............................. ..................
3.4 BED RECOVERY .................... ..............................
12
12
13
18
18
24
27
29
ENERGY DISSIPATION AND ELEMENTS OF MODELING ...............
4.1 WAVE ENERGY DISSIPATION OVER NONRIGID BEDS .........
4.2 A MODELING PERSPECTIVE .
. . .. 31
. . ..... 31
a a a . . ...................................... 31
4.3 BASIC CONSERVATION EQUATIONS .............
4.4 CONSTITUTIVE EQUATIONS .............. .....
4.4.1 Fluid(incompressible) ....................
4.4.2 Elastic Material .........................
4.4.3 Viscoelastic Material ....................
4.5 MODELS FOR WATER WAVES OVER MUD (XZ PLANE)
4.5.1 Inviscid, Shallow Water over Mud .............
4.5.2 Viscid Fluid (Water) over Viscid/Viscoelastic Mud .
4.6 MODELING WATER COLUMN SEDIMENT DYNAMICS .
LABORATORY/FIELD DATA AND MODEL SIMULATIONS ......................... .....
5.1 INTRODUCTION ............................. ....................
5.2 WAVE FLUME RESULTS ...........................................
5.2.1 Test Facility, Experiments and Modeling Approaches .....................
5.2.2 Wave Attenuation and Mud Motion ...................................
5.2.3 Bed/Interface Erosion ..............................................
5.3 LAKE OKEECHOBEE, FLORIDA .......... ............................ .
5.3.1 Setting .................... .................................
5.3.2 M ud Motion ............................ ...................
5.3.3 Resuspension ...............................................
5.4 SOUTHWEST COAST OF INDIA .......................................
5.5 MOBILE BERM, ALABAMA ............ ...............................
33
34
34
34
34
36
36
40
42
48
48
48
48
51
56
59
59
60
62
62
65
BIBLIOGRAPHY ...................................... .. ........................ 70
APPENDICES
A EXPLANATORY NOTES FOR TABLE 7 ............. ...... ............... 77
B SUMMARY OF SOME WAVEMUD MODELING STUDIES ......................... 84
0 0 a 0
0 0 e 0
LIST OF FIGURES
Figure Page
1 Schematic description of the processes and feedbacks related to the response of mud bed to
water waves. Box with dashed line represents the mudwater system, inverted triangle
represents forcing, rectangles are components of the mudwater system, ellipses are process
transfer functions and circles represent manifested responses (adapted from Maa and Mehta,
1989).... ... .. ......... ... ... ... ... .. ... . ... .. .. 9
2 Schematic diagram showing the vertical variation of the density of mudwater mixture
in the wave environment. MWL = mean water level . . . .... ........ 14
3 a) Settling flux variation with excess density for sediment from the Severn Estuary,
United Kingdom. Laboratory column data reported by Odd and Rodger (1986); b) An
instantaneous excess density profile showing a marked lutocline in the Severn
Estuary, United Kingdom (data reported by Odd and Rodger, 1986) ................ 14
4 Shear rigidity modulus versus solids volume fraction for sediment S1 (after James et al.,
1988) ...................................... ................. 16
5 Schematic of timevariation of pore pressure under wave action .................... 18
6 Schematic of instantaneous stress profiles in a watermud system .................. 19
7 Timevariation of effective stress at three elevations (above flume bottom) in a clay bed
kaolinitee + attapulgite) subjected to 1 Hz progressive waves of 4 cm amplitude (after
Feng, 1992) .................. ........................... ...... 19
8 Timevariation of the depth of fluidization in three tests with different wave amplitudes
and consolidation periods. Data points from Feng (1992) ...................... 21
9 Development of a fluid mud layer starting with a "bed" of Hillsboro Bay mud (Run 1; see
Ross and Mehta, 1990). df is the vertical distance between the lutocline and the a' = 1 Pa
level. Note the occurrence of an initial, 1.1 cm thick layer of fluid mud over the bed, as a
result of a local disturbance of the bed prior to test initiation. The observed initial increase
in the lutocline elevation is attributed to advection of fluid mud to the measurement site .... 21
10 Mud density profiles from Run 1 (Fig. 9). Arrows indicate bed level (a' = 1 Pa) at
different times at test initiation (level 1), and following test initiation (after Ross and
Mehta, 1990) ................................ .................. 23
11 Timevariation of relative shear wave velocity in a bed of attapulgite + kaolinite of 20 hr
consolidation subjected to a 1 Hz progressive water wave of 2 cm forcing
amplitude (after Williams and Williams, 1992) ................ .............. 23
12 Storage modulus, G'(9), and loss modulus G"(o), versus strain amplitude, y, for K2 kaolin
at a salinity of 35 ppt and a solids concentration of 41%. w = 1.5 rad/s. Lines are drawn to
enhance data trends (after Chou, 1989) ................................... 25
13 Contours of storage modulus, G' in (a) and loss modulus, G", in (b) for K1 kaolin at a
salinity of 35 ppt (after Chou, 1989) ....................................... 25
14 A fourlayer viscoelastic model (adapted from Chou, 1989) . . . . .. 26
15 Thickness of fluid mud (for a clayey sediment) as a function of wave height, H, assuming a
water depth of 5 m, a wave frequency of 0.1 Hz (adapted from Chou, 1989) ........... 26
16 Influence of waves on shear resistance to erosion of kaolinite beds in flumes (after
Mehta, 1989) ..................................................... 29
17 Normalized rate of erosion versus normalized excess shear stress for a bay mud (after
M aa and M ehta, 1987) ...................... ...... .................. 30
18 Threeelement viscoelastic models: a) Voigt element (represented by 2A2, 2G2) modulated
by additional elasticity; b) Maxwell element (represented by 2/2, 2G2) constrained by
elastic modulus represented by 2G1 ...................................... 37
19 Watermud system in the xz plane ........................................ 37
20 Variation of with normalized time, i (after Cervantes, 1987). . . . . ... 43
21 Wavemud interaction flume (after Maa, 1986) . . . ..... .. ......... 43
22 Diffusion flux, Fd, as a function of apD/az for a6=2 and 06=4.17 (modified from Ross,
1988) ...................................... ................. 46
23 Settling velocity (and flux) variation with pD. Data points obtained from tests using a
lake (Okeechobee, Florida) sediment in a laboratory column (after Hwang, 1989) ........ 47
24 Settling velocity versus pD data using a bay (Townsville, Australia) sediment under two
conditions: quiescent water and water agitated by vertically oscillating rings (adapted
from W olanski et al., 1992) ............... .......................... 47
25 a) Timevariation of the bottom mud density profile during an erosion experiment (Run 1)
using mud from Cedar Key, Florida (after Maa, 1986); b) Uniform density profile of a
mixture of an attapulgite and a kaolinite (after Jiang, 1993) . . . . .. 49
26 Magnitude of equivalent dynamic viscosity (applicable to the viscoelastic model of
Fig. 18a) as a function of wave forcing frequency and mud volume fraction, 4,
forM B mud .................................................... 51
27 Wave spectra from synchronous wave stations 22, 11 and 4 km offshore, showing
loss of wave energy and change in the shape of the spectrum over a fluidlike mud bottom
along the central Surinam coast. Hms = root mean square wave height, T = wave period
and h = water depth (after Wells and Kemp, 1986) ........................... 52
28 Wave attenuation coefficient against frequency for two AK mud (4=0.16) depths,
12 cm and 18 cm. Water depth was 16 cm. The input wave amplitude was 1.3 cm. Circles
are data; lines are simulations based on the model of Jiang (1993) ................. 53
29 Comparison between measured and modelcalculated profiles of the amplitude of
horizontal acceleration in a watermud (AK) system. The water depth was 16 cm, mud
thickness 18 cm, water surface forcing wave amplitude 0.5 cm, wave frequency 1 Hz and
0=0.12 (after Jiang, 1993) ............................................ 53
30 Mass transport velocity in AK mud. Wave amplitude 2 cm, frequency 1 Hz, hi = 14 cm
and mud thickness h2 = 17 cm (after Jiang, 1993) ........................... 55
31 a) Horizontal and vertical velocity amplitudes in water and CK mud; b) Dynamic
pressure amplitude profile corresponding to Fig. 3 la. Smooth profiles have been drawn based
on spatially discretized calculations (after Man and Mehta, 1987; Man, 1986) ........... 55
32 Timevariation of normalized dry density (concentration), D(t) = PD(t)Ds, for CK
mud, during test T1. Mud depth (h2) was 16 cm, water depth (h1) was 17 cm, forcing
wave amplitude (a,) was 3 cm and frequency (f) was 1 Hz (after Cervantes, 1987) ....... 57
33 Phases in the timevariation of PD(t), such as in Fig. 32 (after Cervantes, 1987) ......... 57
34 Measured and modelcalculated dry density (concentration) profiles during erosion of
Hillsboro Bay mud (HB). Water depth (hj) 31 cm, mud depth (h2) 12 cm, forcing wave
amplitude, a, = 3 cm and wave frequency, f = 1 Hz (after Ross, 1988) .............. 58
35 Vertical suspended sediment dry density (concentration) profiles obtained before, during,
and after the passage of a winter cold front at a coastal site in Louisiana. Times are
relative to time of measurement of the prefrontal profile 1 (adapted from Kemp and
W ells, 1987) ................................................... 58
36 Mud thickness contour map of Lake Okeechobee, Florida (after Kirby et al., 1989) ........ 59
37 Typical mud bottom density profile based on vibrocore data from two nearby sampling
sites (after Hwang, 1989). .......................................... 60
38 Data and model application, Lake Okeechobee, Florida: a) Measured water wave
spectrum; b) Comparison between measured and simulated water velocity spectra;
c) Comparison between measured and simulated mud acceleration spectra (after Jiang,
1993) ... ..... ....... .......... ....... ... ..... .. .... .... ..... 61
39 a) Simulated timeevolution of suspension profile due to 0.9 m high, 4 s waves in a
4.6 m deep water column; b) Settling of sediment once waves cease 11 hr from start
of wave action (after Hwang, 1989) ..................................... 63
40 Coastal site off Alleppey in Kerala, India, where monsoonal, mudbanks occur. The
pier is 300 m long (after Mathew, 1992). .................... .............. 64
41 Schematic profile of mudbank off the coast of Kerala, India (after Nair, 1988) .......... 64
42 Offshore and inshore wave spectra off Alleppey: a) without mudbank; b) with mudbank (after
M athew, 1992) ................................................. 65
43 Comparison between measured and modelsimulated inshore wave spectra off Alleppey
in Kerala, India: a) fair weather condition (mudbank absent); b) monsoonal
condition (mudbank present) ......................................... 66
44 Construction site (2,750 m long corridor for dredged material placement) for the Mobile
berm, and offshore/inshore sites of wave measurement (after McLellan et al., 1990) ....... 67
45 Offshore and inshore wave spectra at the Mobile berm site for two different wave
conditions at the offshore site characterized by the maximum wave height, Hx:
a) Hax = 0.9 m; b) H = 1.5 m (after McLellan et al., 1990) .................. 68
46 Comparison between measured and modelsimulated inshore wave spectra at the Mobile
berm site corresponding to two offshore wave conditions: a) IHIm = 0.9 m (see Fig. 45a);
b) Hmax = 1.5 m (see Fig. 45b). Measured offshore spectra have been included for reference 69
LIST OF TABLES
Table Page
1 Some issues in which application of knowledge of mudwave dynamics plays an important
role ...................................... .................. 10
2 Selected definitions of mud ....................................... ...... 12
3 Fluid mud density and corresponding solids volume fraction ranges ................. .16
4 Fluidization depth related parameters ................ .... ............... 20
5 Laboratory based, waveinduced mud erosion rates ................. .......... 28
6 Bed recovery in Test No. 9 of Feng (1992) ................................ 30
7 Some energy dissipation rate expressions due to nonrigid bed mechanisms ............ 32
8 Test parameters including wave frequency w and nominal wave amplitude, a0, and
coefficients 3o, 34 and 61 for empiricalfit equation in Fig. 20 ................... 43
9 Coefficients of Eq. 75 for MB mud ...................................... 51
SYNOPSIS
The basics of the interaction between progressive water waves and a compliant mud bed in the shallow coastal
environment have been briefly reviewed. A key feature of the interactive process is the ability of waves to fluidize
bottom mud and sustain it in that state as long as wave action continues. Compliant or fluid mud is a highly viscous
medium which can oscillate as waves pass over, and cause wave heights to attenuate significantly. At the same time,
given a high enough fluid stress at the mudwater interface particulate entrainment can occur, thus increasing water
column turbidity.
The wavemud interaction problem, involving the prediction of surface wave attenuation, bottom mud motion
and interfacial entrainment or erosion, is simply treated here as one primarily concerned with vertical exchanges
of momentum and sediment mass. A dichotomy inherent in such a treatment arises from the need to assume mud
to be a continuum when simulating wave attenuation and mud motion, while considering the vertical transport of
sediment in the water column to be a twophased problem amenable to classical approaches in sediment transport.
At the center of the continuum approach is the requirement to describe the constitutive relations characterizing
mud rheology. Depending on the properties of the constituent sediment and the ambient fluid, mud in general can
range from being a highly rigid and weakly viscous material to one that can be approximated as a purely viscous
fluid. In that context, fluid mud is better defined as fluidlike mud, since it is not always wholly devoid of rigidity.
As wave action proceeds, the viscoelastic (or poroelastic) properties of mud tend to change with time over
scales that can be two to three orders longer than the typical wave period. The resulting feedbacks in terms of the
influence of this change on wave propagation are not always easy to quantify. Yet, over the past three to four
decades simple hydrodynamic models have been developed to simulate wave attenuation, mud motion and interfacial
erosion in the prototype environment with an acceptable degree of accuracy.
Conventionally, model applications have mainly dealt with investigations of natural phenomena including
predictions of coastal wave heights and turbidity generation for understanding beach erosion and flooding, planning
of port and harbor facilities including safe navigation channels, and design of offshore structures where bottom
stability may be a problem. In recent years applications have been extended to problems of water quality and benthic
biota, since both can be influenced by mud motion and associated constituent fluxes. Most recently, engineering
interest has grown in beneficially using dredged finegrained material from navigation channels to create offshore
underwater berms that can absorb wave energy and thus act as buffers against wave attack in areas in the lee of the
berms. To that end, applications of wavemud interaction modeling have been carried out to simulate the degree
of wave damping that will occur for a berm of given dimensions and mud rheology. Thus, in turn this type of
modeling can be potentially useful in developing guidelines for designing wave energy absorbing berms in future.
INTRODUCTION
1.1 MUDWAVE DYNAMICS IN ENGINEERING AND SCIENTIFIC APPLICATIONS
Advancements in numerous scientific and engineering applications in the shallow marine environment are
contingent upon a better understanding of the physical processes and associated feedback mechanisms that are
inherent in the dynamical response of a muddy bottom subjected to wave action. Some relevant areas and their
descriptions are briefly noted in Table 1. They highlight the wide ranging scientific and engineering concerns in
which mud dynamics plays a key role, including coastal fisheries and agriculture, shore and channel protection, oil
and gas exploration, water quality, shallow marine habitats and so on. Despite the diversity and complexity of the
physical, physicochemical and biological processes involved in these situations, in most cases the processes and
feedbacks related to the physics of mudwave interaction can be identified in a simplified manner, and is the focus
of this report.
1.2 PROCESSES AND FEEDBACKS
The response of a mud bed to wave forcing is schematized in Fig. 1. A brief description of the numbered
pathways is as follows: 1,2) water waves determine the flow field; 3,4) flow field and bed properties together
govern the character and the dynamics of the interface; 5,6) wave loading, consolidation, fluidization and thixotropy
Figure 1. Schematic description of the processes and feedbacks related to the response of mud bed to
water waves. Box with dashed line represents the mudwater system, inverted triangle represents
forcing, rectangles are components of the mudwater system, ellipses are process transfer functions and
circles represent manifested responses (adapted from Maa and Mehta, 1989).
Table 1. Some issues in which application of knowledge of mudwave dynamics plays an important role
Problem Area
Description
Fisheries
Shore protection
Agriculture
Oil and gas exploration
Port navigation
Eutrophication and other water
quality issues
Habitat protection, restoration and
enhancement
Oil spill
Natural coastal mud banks provide a relatively wavefree environment
and where, due to nutrientrich mud, fish catch is often high, e.g. off
Surinam and India (Wells, 1983; Nair, 1988; Mathew, 1992).
Engineered underwater berms in the nearshore region can substantially
reduce wave height as waves pass over the berm towards the shore or a
navigable channel, e.g. off Mobile Bay, Alabama (McLellan et al.,
1990; Dredging Research Technical Notes, 1992).
Coastal mud levees and dikes are designed to reclaim land for
agricultural purposes, e.g. in Hangzhou Bay, China (Wang and Xue,
1990).
Offshore platforms must be designed to withstand failure due to
episodic mud slides and mud fluidization, e.g. in the Mississippi Delta
area (Sterling and Strohbeck, 1973).
Channel depth based on the "nautical depth" concept, in which
underkeel clearance is determined on the basis of mud resistance to
ship motion (Migniot and Hamm, 1990). Design of underwater sills to
prevent port channel shoaling (Tsuruya et al., 1990).
In these problems in lakes, reservoirs and coastal waters, it is desirable
to understand/control turbidity generation due to bottom mud
resuspension (Somly6dy et al., 1983; Mehta, 1991a). Scavenging of
waterborne contaminants by suspended finegrained sediment and its
ultimate deposition in quiescent areas can, in fact, lead to an
improvement in water quality. In such a case, waveinduced turbidity
can therefore be generated, at least in principle, to accomplish
contaminant scavenging, as often occurs naturally in bays and estuaries
(e.g. Krone, 1979).
Design of stable habitats may be accomplished by appropriately shaping
the bank to afford greatest protection against wave action, a desirable
feature in coastal wetland creation (e.g. Conners et al., 1990; National
Research Council, 1994).
Tracking the fate of coastal oil spills requires predictive process models
for the entrainment of oil emulsion in mud and release of oil from mud
into wavegenerated waters (National Research Council, 1989).
erosion; 11,12) shear (and normal) stresses together with mud properties determine mud motion; 13,14) mud change
mud properties with time; 7,8) flow field and interfacial character determine the interfacial shear stress; 9,10)
interfacial shear stress and interfacial properties determine the rate of particle entrainment, or interfacial properties
largely determine the rate of surface wave damping or attenuation.
In this report we will examine the interrelationship between forcing and responses identified in Fig. 1, and
in so doing inherently highlight significant feedbacks represented by some of the numbered pathways. Feedbacks
shown tend to be timedependent, typically varying the behavior of the system gradually in comparison with the
forcing period, as the mud properties change under continued wave action. Thus, predictive approaches for
determining wave attenuation, mud motion and erosion are dependent on a knowledge of the constitutive properties
of mud, which themselves vary widely depending on the mineralogical composition of mud, and modulation of
inorganic properties by biochemical and biophysical influences. Consequently, a few important tests for
characterizing mud properties and transport, e.g. erodibility, must accompany any predictive effort. However,
notwithstanding sitespecificity that is inherent in every problem, an attempt has been made in the report to
emphasize those physical principles that are common to most problems.
The degree of importance of a particular response of the mudwater system evidently depends on the problem
area and the application sought. Thus, for example, wave attenuation is of primary significance for mitigation of
shoreline erosion and coastal flooding. Mud motion determines bottom stability, hence the integrity of structural
foundation. It also controls the intake and release of nutrients and contaminants across the mudwater interface.
Finally, coastal and estuarine turbidity due to wave action is the result of particulate entrainment at the mud/water
interface. In any event, as a first step it is essential to introduce basic definitions related to the physical state of mud
in the wave field. This aspect is considered next.
MUDWAVE DOMAIN
2.1 DEFINITIONS
Three previous definitions of mud are given in Table 2, two ocean science related and the third from coastal
engineering. From the present standpoint, the coastal engineering description is perhaps more relevant than the other
two because it refers to the state of mud (fluidtoplastic) as opposed to composition alone. In that context it is
noteworthy that wave attenuation and mud motion are both dependent on the constitutive relations characterizing
mud rheology, which in turn depends on the physical state of mud. On the other hand, particle entrainment at the
interface largely depends on the structure of the particulate aggregate network, which is specified by sediment
composition for a given composition of the fluid (ionic species in water, their concentration, and pH). Mud that is
predominantly composed of particles smaller than about 20 /m is much more cohesive than one in which the
particles are coarser (Mehta and Lee, 1994). Where it is essential to make this distinction, we will refer to the
former as cohesive mud.
Table 2. Selected definitions of mud
Source Field Definition
Hunt and Groves (1965) Ocean science Pelagic or terrigenous detrital material consisting of
particles smaller than sand, i.e., an undifferentiated
sediment made up of particles mostly within the
siltclay range smaller than 0.0625 mm.
Allen (1972) Coastal engineering A fluidtoplastic mixture of finely divided particles
of solid material and water.
Tver (1979) Ocean science Pelagic or terrigenous detrital material consisting
mostly of silt and claysized particles (less than
0.06 mm), but often containing varying amounts of
sand and/or organic materials. It is a general term
applied to any sticky finegrained sediment whose
exact size classification has not been determined.
The forcing frequency (f) of common interest relative to gravity waves ranges widely, from about 10 Hz to
105 Hz. The former corresponds to the transition whereby surface tension becomes increasingly important as a
restoring force with increasing frequency, while the latter is representative of the frequency of the astronomical tide.
It should be added, however, that frequencies that are lower than tidal, especially those corresponding to subtidal
oscillations having periods on the order of days, can also be important in some situations, for example with regard
to long term erosion or accretion of finegrained sediments in windforced bays.
Given water depth, h, and particle settling velocity, ws, two characteristic, frequency dependent numbers that
together characterize the primary nature of wave forcing and bottom response are hw2/g and hw/w,, where w=2rf
and g is the acceleration due to gravity. When the first number is less than 0.3, fluid pressure is practically
hydrostatic, and the celerity of the shallow water wave, = (gh)1/2. As ho2/g exceeds 0.3, dynamic pressure effects
become increasingly important with increasing water depth, or frequency. The second number scales the water depth
in relation to the settling velocity for a given frequency. For finegrained sediment, selecting w,= 104 m/s, h= 10
m and c= 1 rad/s as characteristic values yields hw/w,= 105, which is up to two orders of magnitude greater than
the corresponding value for a sandy bed, for example. A manifestation of this difference is related to the vertical
structure of density, which assumes considerable importance when the sediment is finegrained. Sedimentinduced
stratification characteristically occurs in this case irrespective of the value of hw2/g, as long as the wave is not in
deep water (h/wl> 7r), since in the latter case wavebottom mud interaction ceases.
Other parametric approaches have also been proposed for characterizing the wavemud environment in
contrast to wavesand environment. For example, McCave (1971) has defined a "wave effectiveness parameter" as
the product of the theoretical instantaneous sediment transport rate times wave frequency. In the southern North Sea
area examined, relatively low values of this parameter correlated with the occurrence of mud.
2.2 FLUID MUD
A simple description of the mudwave system is given in Fig. 2, in which um is the amplitude of the
horizontal wave orbital velocity. The (bulk) density of the watermud mixture, p, varies from p, at the surface to
Pb at the bottom. The quantity p, is water density. A sharp density gradient, or lutocline (Parker and Kirby, 1982),
separates the upper column suspension from fluidized, compliant mud below. At the bottom of the mobile
suspension the density rises from pu to pt, which is the range over which fluid mud is considered to occur. The bed
below fluid mud can undergo deformation. This deformation in fact may eventually break the interparticle or inter
aggregate bonds, and thus change the bed, possessing a structured matrix, to fluid mud. Below the level at which
the depth of penetration of the wave orbit practically ends the bed remains uninfluenced by wave motion.
Fluid mud characterizing densities, pu and pt, are operationally defined as follows. The flux of sediment
settling within the mobile suspension, F,=wsp,, where 0=(pp,)/(p,pw) is the solids volume fraction (=ln,
where n is the porosity) and p, = granular density of the sediment. In Fig. 3a, data from a laboratory settling
column using sediment from the Severn Estuary in the United Kingdom have been used to plot F, against the excess
density, Ap=pp, (adapted from Odd and Rodger, 1986). The peak value of F,=Fnm=40 g/m2*s corresponds to
the onset of hindered settling, under which F, decreases with increasing Ap (Mehta, 1990). Hindered settling begins
at Apm=pmipw=9 kg/m3, where pm is the value of p when Fs=Fsm. We further note that at the peak flux,
aFs/aAp=0. Furthermore, given z as the vertical coordinate, aFslaAp=(aF,/Iz)/(aAp/az). Thus, since at the
lutocline aAp/az tends to infinity the condition, aF/OAp =0, is satisfied there (horizontal line in Fig. 2), irrespective
of the value of aFs/8z. Thus, in reality Apm practically coincides with Apu=ppu In Fig. 3b the correspondence
between the lutocline elevation, zu, and Apu is shown for a measured density profile from the Severn. Note that the
sediment remains suspended in this estuary largely as a result of tidal flow, as opposed to wave action. Where waves
predominate, the essential description should remain unchanged, however.
Figure 2. Schematic diagram showing the vertical variation of the density of mudwater mixture in the wave
environment. MWL = mean water level.
EXCESS DENSITY, Ap (kg/m3)
102 0 10 20 30 40 50
EXCESS DENSITY, Ap (kg/m3)
Fig. 3. a) Settling flux variation with excess density for sediment from the Severn Estuary, United Kingdom.
Laboratory column data reported by Odd and Rodger (1986); b) An instantaneous excess density profile showing
a marked lutocline in the Severn Estuary, United Kingdom (data reported by Odd and Rodger, 1986).
Lutoclines have been found to persist even under fairly drastic waveinduced agitation, as a result of the
significant negative buoyancy of bottom mud. However, under highly eroding conditions, p, may decrease
measurably below pm (Ross and Mehta, 1989) and, in fact, the equality between Pm and Pu is valid only for
conditions under which the sediment settles without resuspension or upward diffusion. It holds reasonably well for
weakly eroding conditions, however.
Fluid mud is better described as "fluidlike" mud, in which the particles are largely (but not always solely)
fluidsupported (Smith and Kirby, 1989). Thus, in general, it is a "quasisuspension". Since the bed below it is
essentially particlesupported the density, p(, is the value of p below which intergranular contact is marginal. Since
if left at rest fluid mud will dewater to form a bed, its existence in the present case is dependent on wave agitation,
which prevents a reduction in the pore pressure, and thereby, in turn, prevents the constituent particles from
developing permanent electrochemical bonds. In many shallow, wavedominated environments, continued wave
action causes the top layer of mud to remain fluidized. For instance, in Lake Okeechobee, Florida wind waves
persistently sustain the top 520 cm thick, organicrich mud in the fluidized state (Kirby et al., 1989; Hwang, 1989).
Since the fluid mud/bed boundary is typically very dynamic, and pore pressures difficult to measure in the
field, practical definitions have been used to determine Pl. One such definition is based on the approximate empirical
relationship between the vane shear strength of the soil, Tr, and p or, equivalently the solids volume fraction, 0:
Ty= ao(oto) where 4f is solids volume fraction corresponding to Pt, and CaI and 0o are sedimentspecific
coefficients that must be determined experimentally. Thus, when 0= 01, 7Y=0, and shearometric evidence has been
used to characterize 01 as the critical, spacefilling solids volume fraction (James et al., 1988). For the mud from
Lake Okeechobee, Hwang (1989) obtained (given 7r in Pa) aco=22.6, (o= 1 and O= 0.06; the latter corresponding
to P, = 1,065 kg/m3. Other definitions of a similar nature, but ones in which T7 is substituted by the upper Bingham
yield stress, ry, which is obtained for the typically pseudoplastic stressrate of strain curve for mud at low rates of
strain, have also been used widely. See Mehta (1991b) for a brief review of these definitions. In recent years a
revised interpretation of Ty as the critical stress at which plastic yield occurs in a creep test has been proposed
(James et al., 1988; Jiang, 1993). This test, which must be conducted in a controlledstress rheometer, provides
a direct measure of 7y, thus obviating approximations inherent in the estimation of Ty.
The most commonly considered values of pu and pg are 1,030 and 1,300 kg/m3, respectively, although other
ranges have been reported, as noted in Table 3 (amended from Ross et al., 1987). The main reason for the observed
variations in p, and pi is that they were determined under different hydrodynamic conditions and for different muds.
Cohesive muds having concentrations 0 > 0t tend to exhibit a measurable viscoelastic response to wave
forcing of stresses below the critical stress for plastic yield. The viscosity, u, and shear modulus of elasticity, or
the rigidity modulus, G, of typical surficial muds found in the shallow marine environment range from 0(101) to
O(104) Pa.s and Pa, respectively. These ranges indicate that such muds tend to be highly viscous but only weakly
to moderately elastic.
By way of an elaborate but also theoretically more accurate definition than stated above, Foda et al. (1993)
consider the bed to be an elastic medium, and "fluid" mud to incorporate a transitional viscoelastic layer between
Table 3. Fluid mud density and corresponding solids volume fraction ranges
Investigators) pu" au p1 a a
(kg/m3) (kg/m3)
Inglis and Allen (1957) 1,030 0.018 1,300 0.182
Krone (1962)b 1,010 0.0061 1,110 0.067
Wells (1983) 1,030 0.018 1,300 0.182
Nichols (1985) 1,003 0.0018 1,200 0.121
Kendrick and Derbyshire (1985) 1,120 0.073 1,250 0.152
Hwang (1989) 1,002 0.047 1,065 0.060
aConversion between density and solids volume fraction by using sediment granular density, p, = 2,650
kg/m3, and water density = 1,000 kg/m3, except for Hwang (1989), in which case p, = 2,140 kg/m3.
bExclusively based on laboratory data. The other definitions rely on field evidence, although they are not
necessarily based solely on field data.
the bed and a viscous layer above. Returning however to the more simple definition, as shown in Fig. 4, typical
shearometric data suggest of to be in the range of 0.05 to 0.1, below which G is comparatively small (James et al.,
1988). 4~ values in Table 3 on the other hand are generally higher, with the exception of Krone's (0.067) and
Hwang's (0.060), both of which are based on laboratory data. The other four are fieldbased, and technically
somewhat arbitrary. At the same time however, any definition of ft based on G (Fig. 4) would also be somewhat
arbitrary, unless it is contingent upon a quantitative criterion concerning the magnitude of G for defining 4(.
Since the dissipative role of mud is largely confined to the top compliant layer (Foda, 1989), predictive
hydrodynamic models that assume cohesive mud to be a purely viscous fluid seem to yield simulations of surface
wave attenuation of acceptable accuracy for field application (Jiang and Mehta, 1992). In other simulations an elastic
1500
1000
Figure 4. Shea rigidity modulus versus solids volume fraction for sediment S (after James et al., 1988).
500
0 0.05 0.10 0.15 0.20
Figure 4. Shear rigidity modulus versus solids volume fraction for sediment S1 (after James et al., 1988).
component has been added (Jiang, 1993). Less dissipative and more elastic silty, "fluidlike" muds have been
considered to be poroelastic (Yamamoto, 1983). In any event, it is essential to estimate the thickness of the fluid
(or fluidlike) mud layer as a precursor to hydrodynamic modeling. The simplest way to identify layer thickness
is in terms of densities pu and pt. However, this very approximate approach does not explicitly account for the
dynamic nature of the fluid mud boundaries, dependent as they are on the nature of wave forcing. Processes
contributing to the generation of fluid mud from bed, and the scour of the fluid mudwater interface leading to its
entrainment are briefly considered next.
GENERATION AND EROSION OF FLUID MUD
3.1 FLUIDIZATION
Under progressive waves that are not in deep water, gradients in pore pressure can cause the pore fluid to flow
relative to the soil skeleton, which may eventually lead to rupturing of the interparticle cohesive bonds, hence a
loss of effective stress. Starting with a bed at rest, the increase in (waveaveraged) pore pressure, u,, with time
at a fixed position is shown schematically in Fig. 5, in which ah = hydrostatic pressure and Au = waveaveraged
excess pore pressure. This waveaveraged effective (normal) stress, a', decreases with a buildup of excess pore
pressure until soil liquefaction occurs when the total stress, a=u.=Au+oh, and a'=O (see, e.g. Lamb and
Whitman, 1969). The waveaveraged portrayal of pore water variation is an evident approximation of a phenomenon
that actually occurs within a twophased medium, in which properties of the sedimentary matrix vary spatially at
the microfabric scale, even when the medium is of uniform bulk density and composition. In fact, careful
measurements of pore pressure oscillations under waveforcing in silty soils suggest that transitional, resonant
amplification of the pressure amplitude accompanies an episodic increase in the wavemean pore pressure at
preferential sites (cavities) within the soil mass (Foda et al., 1991). A somewhat similar occurrence is reported in
a clayey soil (Feng, 1992). An important implication of this observation is that at the microfabric scale, soil
liquefaction is initiated in a spatially and temporally inhomogeneous manner.
An instantaneous view of the stress profiles in the mudwater system is shown schematically in Fig. 6. When
the initial bed matrix is disturbed by wave motion, the general manner in which fluidization, when considered
synonymous with liquefaction, occurs depends on several factors, an important one being the thickness and the
degree of consolidation of the mud layer. When the thickness is small and the mud is unconsolidated or partially
consolidated, mud displacement tends to be greater near the rigid bottom below compliant mud, than near the
surface. In this case fluidization can conceivably proceed from the rigid bottom up. In a comparatively deep mud
layer, or a consolidated thinner layer, the displacement is greatest at the top and decreases with depth; hence in this
/ A
I,
O0 l' O
TIME
Figure 5. Schematic of timevariation of pore pressure under wave action.
z
0
F \Fluid Mud Surface (Lutocline)
5 \ = Fluid Mud
> FuBed Surface
I
\ Bed
> \U
Au 0 ,
STRESS
Figure 6. Schematic of instantaneous stress profiles in a watermud system.
80 Elev. 12 cm
40 1
l  o I  I  I 
TIME (min)
Figure 7. Timevariation of effective stress at three elevations (above flume bottom) in a clay bed kaolinitee +
attapulgite) subjected to 1 Hz progressive waves of 4 cm amplitude (after Feng, 1992).
case fluidization tends to proceed from the top (Chou, 1989). Formally then it appears that the manner in which
fluidization occurs characteristically depends on the relative water depth, kh1, and the wave Reynolds number in
2
mud, wh2b/2, where k = wave number, w = wave frequency, hi = water depth, h2 = mud depth and v2 =
kinematic viscosity of mud. For small values of khl and wh22/r2 bottomup fluidization should be favored. For
relatively large values of these parameters, the reverse process should be facilitated, provided there is sufficient
wave energy input to initiate and sustain fluidization. Imposing a nonuniform bed density gradient, with density
increasing with depth, should further enhance the likelihood of fluidization occurring from top, since in this case
the dynamic pressure gradient tends to decrease rather markedly with depth (Maa and Mehta, 1987).
When fluidization proceeds downward from the mud/water interface an equilibrium thickness of the fluid mud
layer is attained. In Fig. 7 the fall in the waveaveraged effective stress with time at three selected elevations where
timeseries of total and pore pressures were measured below the interface is shown from a preliminary test (No.
10; see Feng, 1992) in a wave flume. The 16 cm thick bed was composed of an aqueous mixture of a kaolinite and
an attapulgite in equal proportions by weight, having a mean density of 1,170 kg/m3. The water depth was 19 cm,
and the 1 Hz forcing wave had an amplitude of 4 cm. The bed was allowed to consolidate for 85 hours before test
initiation. Notice the increasing time required for fluidization with depth below the interface.
The thickness of the fluid mud layer, d(t), in general can be expressed as
df = dfo, f(t) (1)
where df, is the equilibrium value of df, such that the function f(t) 1, as t * oo. Data such as those shown in
Fig. 7 were used to determine the increase in df with time in Fig. 8 from three tests (Nos. 8, 9 and 10) summarized
in Table 4. The comparatively short durations of these tests were insufficient for the attainment of equilibrium
depths, dfe. In any event, selecting the following empirical equation for fitting the data points,
df = MI[1 e it, ]2.5 (2)
Table 4. Fluidization depth related parameters
Wave
S.ae Consolidation Period MI
Test No. Amplitude (hr) (cm)
((cm)Or) (cm)
(cm)
8 2.0 240 18
9 2.8 65 50
10 4.0 85 35
E
3 10
W
0
Z
0
5
L.
U.
30000
TIME, t (s)
Figure 8. Timevariation of the depth of fluidization in three tests with different wave amplitudes and consolidation
periods. Data points from Feng (1992).
61 I I I
0 0.5 1.0 1.5
TIME (hr)
Figure 9. Development of a fluid mud layer starting with a "bed" of Hillsboro Bay mud (Run 1; see Ross and
Mehta, 1990). df is the vertical distance between the lutocline and the a'=1 Pa level. Note the occurrence of an
initial, 1.1 cm thick layer of fluid mud over the bed, as a result of a local disturbance of the bed prior to test
initiation. The observed initial increase in the lutocline elevation is attributed to advection of fluid mud to the
measurement site.
we note that the curves drawn are obtained by retaining constant values of the coefficients 1l(=0.0216) and
31(=0.346) for the timedependent function [f(t)] within the brackets, while varying M1 (Table 4). Thus the
influences of wave amplitude and bed consolidation period are contained wholly in M1. Since both the amplitude
and the consolidation period were varied in each test their individual effects cannot be gleaned; however we note
that the trend of increasing rate of fluidization corresponds with decreasing bed consolidation. Thus the initial state
of the bed seems to have had a dominant influence on the rate of fluidization in these particular tests. Note that
according to the form of Eq. 2, MI=df,; yet, because dfe was not actually attained, M1 should preferably not be
set equal to df, and the applicability of Eq. 2 should be restricted to the experimental durations only.
A difficulty with tracking waveinduced fluidization by means other than pore pressures, such as by measuring
accompanying changes in bed density, is that conventional methods for measuring density, e.g. by collecting mud
samples for subsequent gravimetric analysis, are often too coarse for the degree of resolution required to detect the
necessary changes. In fact, there may be no "significant" change, as in the case of the fluidization test shown in
Fig. 9, in which the timevariation of df is shown for a bed composed of a predominantly montmorillonitic estuarine
mud (from Hillsboro Bay, Florida). The water depth was 31 cm and the nominal mud thickness 12 cm. The bed,
which was prepared 168 hr before test initiation, was forced by a 1 Hz wave of 3 cm amplitude. The boundary
between the fluid mud layer and the cohesive bed is defined by the locus of the elevation corresponding to or'=
Pa, a very small value. During the first hour, vertical profiles of mud density were measured (by gravimetric
analysis of samples withdrawn from the side of the experimental flume via tubes), as shown in Fig. 10. Arrows 1
through 4 mark the positions of the a'= 1 Pa boundary in Fig. 9. Note that between levels 1 and 4 no measurable
change in density occurred (Ross and Mehta, 1990).
Sensitive measures of "soil softening" by waves include such lumped parameters as the undrained shear
strength and the shear modulus or rigidity modulus, G, of the bed (Thiers and Seed, 1968; Schuckman and
Yamamoto, 1982). Within the bed of kaolinite and attapulgite mentioned earlier, and having a pretest consolidation
period of 20 hr, a specially designed, miniature shearometer was installed to record the change in G with time under
continued wave action (Williams and Williams, 1992). The device measured the speed of propagation, V, of a
1.8 kHz shear wave generated and detected by transducers incorporating piezoceramic "bimorph" elements bonded
to miniature steel plates which served as the generating/detecting surfaces (length 12 mm; width 4 mm; thickness
100 pmr). At this high shear wave frequency, V was practically instantaneous in terms of its variation over the
experimental time scale. Given the initial speed V(0), the corresponding ratio, V(t)/V(0), is plotted against time in
Fig. 11. Notice the "exponential" fall in the speed, which is commensurate with a corresponding increase in df such
as that observed in Fig. 8. The best fit equation is
V(t) = e/2 (3)
V(0)
where a2=0.0216 and 02=0.346. Recognizing that the forms of the timedependent terms in Eqs. 2 and 3 are
analogous, and that, perhaps fortuitously, al =a2 and P =32, eliminating time between the two equations leads to
E
z
0
w 6
J
4
1
2 II I I
1060 1100 1140 1180
DENSITY (kg/m3)
Figure 10. Mud density profiles from Run 1 (Fig. 9). Arrows indicate bed level (a' = 1 Pa) at different times at
test initiation (level 1), and following test initiation (after Ross and Mehta, 1990).
1.0 I i I I
v (t)
= :exp (a2tP2)
o.9 / V (0)
> V(O) = 2 m/s
a 2 =0.0216
. 0.8 = 0.346
0.7
J
W 0.6
0.5I I I
0 2000 4000
TIME, t (s)
Figure 11. Timevariation of relative shear wave velocity in a bed of attapulgite + kaolinite of 20 hr consolidation
subjected to a 1 Hz progressive water wave of 2 cm forcing amplitude (after Williams and Williams, 1992).
df = M1 V(t) 2.5 (4)
which indicates a seemingly unique dependence of df on V for a given bed. For a given shear wave forcing
frequency, G depends on V as well as on the phase angle between the shear stress amplitude and the resulting
amplitude of strain (Williams and Williams, 1992). Thus the coefficient M1 essentially contains the influence of the
phase angle in an unquantifiable way. Nevertheless, Eq. 4 demonstrates the dependence of df on a soil dynamical
parameter that intrinsically reflects the physical state of mud.
3.2 ESTIMATION OF FLUIDIZATION THICKNESS
Research on the estimation of the fluidization thickness apparently has largely focused on the equilibrium
value, dfe, as opposed to dg(t). Thus, for example, the linkage between mud rheology and dfe was explored by Chou
(1989), by considering bottom mud to be generally viscoelastic. Onset of fluidization was defined in terms of
critical values of shear strain related to elastic and viscous responses of the bed. This criterion is mathematically
different from that for liquefaction (i.e. a' =0). Yet, the two characterize the same phenomenon. As noted in Section
4.4.3, the constitutive equation for a viscoelastic material is, Tji =G*Eji, where Tji = deviatoric component of
stress, Ej = deviatoric component of strain, subscripts ij denote directions and G* is the complex shear modulus.
By definition, G*=G'iG", where G' = storage modulus (associated with the elastic component) and G" = loss
modulus (associated with the viscous component). For a Voigt material, G'=G and G"=/ew, where j = viscosity,
G = shear modulus of elasticity and w = forcing frequency (Barnes et al., 1989).
The behaviors of G' and G" as functions of the amplitude of applied oscillatory strain, y., were examined
by Chou (1989) in a controlledstrain rheometer. Figure 12 shows the results for a kaolin. Note the importance of
elasticity at low strains when viscous loss was practically nil. With increasing strain the clay behaved increasingly
as a viscous material. Results such as these were used to develop the nomograms of Figs. 13a,b for G' and G" as
functions of strain amplitude, y. and relative mud density (p/p,). The forcing frequency was held constant at 1.5
rad/s. However, in experimenting with changing the forcing frequency it was found that the influence of y was
much more pronounced than that of w in controlling G' and G". Depending on the value of y the domain is seen
to be subdivided into three regimes viscous, viscoelastic and elastic; y and Ye are values of the strain, y, that
define the appropriate boundaries.
The G', G" nomograms enable an empirical determination of the thicknesses of the viscous and viscoelastic
layers in an otherwise elastic bed. Chou (1989) considered the watermud system shown schematically in Fig. 14,
and calculated the equilibrium thicknesses of the fluidlike viscous layer and the soft viscoelastic layer as functions
of the wave height H=2a, where a is the forcing wave amplitude. The procedure involved the use of a wavemud
interaction model in conjunction with the G', G" nomograms. For a Voigt viscoelastic mud the model becomes
analogous to that of Maa (1986). The model was used iteratively to calculate strain amplitudes at different depths
102
1I i 1 1 1 II I I 1111 I I
0.001 0.01 0.1 1
STRAIN AMPLITUDE
Figure 12. Storage modulus, G'(9), and loss modulus G"(o), versus strain amplitude, y*, for K2 kaolin at a salinity
of 35 ppt and a solids concentration of 41%. w = 1.5 rad/s. Lines are drawn to enhance data trends (after Chou,
1989).
?
ui 1
1 101
z 102
I
Cf) 103
RELATIVE MUD DENSITY, p/pw
Figure 13. Contours of storage modulus, G' in (a) and loss modulus, G", in (b) for K1 kaolin at a salinity of 35
ppt (after Chou, 1989).
F_ Wave Propagation
Free Surface Wave Length, L
x
Water Column
Mudline 
FluidLike Mud
z=h2
Soft Mud
z = h3
Stiff Mud
Figure 14. A fourlayer viscoelastic model (adapted from Chou, 1989).
1.0
0.8
0.6
I
h2 h1
& 0.4
0.2
0 0.50 1.00 1.50
WAVE HEIGHT (m)
Figure 15. Thickness of fluid mud (for a clayey sediment) as a function of wave height, H, assuming a water depth
of 5 m, a wave frequency of 0.1 Hz (adapted from Chou, 1989).
as functions of the wave amplitude, and hence the depths hi and h2. An example of the fluidized or fluidlike layer
thickness, h2hl, as a function of wave height is shown in Fig. 15.
3.3 SURFACE EROSION
Since the action of waves over a cohesive bed leads to a fairly rapid development of the fluidized mud layer,
and since fluidization can be considered to be a mode of entrainment, it is sometimes reasonable to consider the
interface between fluid mud and the bed to be the appropriate bed "surface". This assumption however becomes
increasingly tenuous with increasing thickness of the fluid mud layer. Accurate models for sediment transport must
therefore consider bed fluidization as a process that is distinct from mud surface erosion. Nevertheless, in a large
number of practical applications the use of stressbased equations for the rate of bottom erosion, akin to those
obtained for unidirectional flow situations, seems to yield acceptable answers with regard to the entrainment of
particles into the water column.
Expressions for the rate of erosion have been derived almost exclusively from laboratory studies. Exceptions
include "back calculated" rates using field data and numerical models for suspended sediment transport (e.g. Sheng
et al., 1986). Some laboratory erosion rate expressions are listed in Table 5. The expression of Alishahi and Krone
(1964) is based on data on the timevariation of the depthmean suspension concentration during erosion reported
by the investigators. Experimental data, in general, typically conform to the relationship
ITRJ
S= M2 lb_1 a3 (5)
where 6 = rate of erosion, rb = peak value of the cyclic bed shear stress, M2 = erosion rate constant, TR =
erosion resistance and a3 = empirical constant. Values of a3 in Table 5 range from 0.95 to 1.82. When a3= 1,
Eq. 5 becomes analogous to the corresponding expression for cohesive bed erosion under steady or quasisteady
flows (see e.g. Mehta, 1988):
S= M3 1 (6)
where M3 is the rate constant and 7, is defined as the erosion shear strength. In the laboratory setting, 7, can be
determined as a function of depth below the bed surface via a procedure involving layerbylayer erosion of the bed
(Parchure and Mehta, 1985). Maa (1986) used a similar procedure to determine TR, and thus, in a sense, established
a correspondence between 7, and TR. Furthermore, the range of 0.15 to 0.39 Pa reported for TR in Table 5 is
generally of the same order of magnitude as that for 7, (see e.g. Mehta, 1988). However, it must also be recognized
that the timechange in the properties of the bottom mud due to wave action tends to influence TR in a significant
way (Maa, 1986). Thus the value of TR used in solving Eq. 5 must be inherently a characteristic timemean value,
unless the variation of TR with time is explicitly included. However, given the usual paucity of data on TR and
Table 3.2. Laboratory based, waveinduced mud erosion rates
Mode of Parameter Values in Erosion Rate
Investigators) Wave Sediment Type Parameter Rangesa'
Generation a (cm); w (rad/s) Expressionb 6 = M2 
Alishahi and Krone (1964) Wind Bay mud 0.9 < a g 3.4 Test 1:
11.6 : w 17.5 M2=0.00048; rR=0.29; a3=1.72
Test 2:
M2=0.0112; TR=0.39; a3=1.15
Thimakom (1980,1984) Mechanical Rivermouth mud (0.16 < ak < 1.60) M2= rb/27rR; TR = variable; a3= 1.00
3.1 & we 12.6
Maa and Mehta (1987) Mechanical Commercial clay, 1.4 < a 3.7 Commercial clay:
Bay mud 3.3 wo 6.3 M2=0.131; TR=f(z); a3=1.15
Bay mud:
M2=0.030; TR=f(z); a3=0.95
Mimura (1993) Mechanical Commercial clay 0.6 < a < 6.9 M2=0.00027; TR=0.15 ; ca3=1.82
4.8 < aw 8.2
'a = wave amplitude; w = wave frequency; k = wave number.
bB = erosion rate (kg/m2s); Tb = peak value of the cyclic bed shear stress (Pa); M2 = erosion rate constant (kg/m2.s); TR = erosion resistance (Pa); 3 =
empirical constant; U = aw/sinh(khl) is the amplitude of the bottom oscillatory velocity, where k = wave number and hi = water depth; bb = (v/2w)1 is
the wave boundary layer thickness where v = kinematic viscosity of water; f(z) = bed shear strength as a function of depth (z); Test 2 (148 hr) had a longer
consolidation period than Test 1 (38 hr).
unquantifiable effects of bed changes on particulate entrainment, the use of timevarying TR is generally not
warranted. It is also noteworthy that, depending on the way in which waves influence the bed, TR may be lower
or higher than r (Mimura, 1993). In Fig. 16, 7R is observed to be lower than r, for a bed of kaolinite.
In Fig. 17, a typical relationship in the form of Eq. 5 based on flume measurements is shown (Maa and Mehta,
1987). Data smearing of the degree observed is typical of such plots, and is at least in part due to the uncertainty
in estimating rR. Furthermore, it is rather selfevident that plots such as these mask influences that may depend on
multiple causes and associated feedbacks. For example, Jackson (1973), who conducted laboratory tests on wave
resuspension of a mixture of silt, clay and some fine sand on a sloping beach, observed rapidly increasing suspended
sediment concentration in the first hour, reaching a maximum after the second hour and decreasing subsequently.
A possible explanation for the decrease in concentration is that the bed became less rigid with time under wave
action, which in turn resulted in greater energy dissipation and reduced the shear stress at the interface. Jackson
also noted that for relatively small orbital velocities (1215 cm/s), there was very little erosion, whereas for
velocities greater than about 20 cm/s the surface erosion process was replaced by mass erosion, with comparatively
much larger rates of sediment entrainment, a behavior also observed under steady flows (Mehta, 1991c).
3.4 BED RECOVERY
Since the experimental data presented in Section 3.2 suggest that the loss of effective stress is primarily due
to the rupturing of interparticle bonds without a significant change in the density or void ratio, it follows that bed
recovery should be comparatively rapid in the laboratory setting after wave action ceases. In test No. 9 of Feng
(1992) mentioned earlier, wave action was maintained for 7.5 hr, following which effective stresses at different
elevations above the flume bottom were determined from pore pressure profiles 4 hr later. Observe in Table 6 that
the percent recovery, as defined, varied from 0 to 77, with a mean of 36. The nonuniformity in percentages
. 0.4
LU o Without Waves (Parchure, 1984)
S* With Waves (Maa, 1986)
00
0.2 I
S /I Wave Effect ..
wc I I
Co 0 '0
O 0 5 10 15
W PERIOD OF CONSOLIDATION (days)
Figure 16. Influence of waves on shear resistance to erosion of kaolinite beds in flumes (after Mehta, 1989).
101
100
o*1 ..*
*4
** 0**
102 0 *
s* __ tbTR 
103 * M TR _
4 I uiiIII I i nil i i I iiiIII I IiiiI
103 102 101 100 101
NORMALIZED STRESS, b TR
TR
Figure 17. Normalized rate of erosion versus normalized excess shear stress for a bay mud (after Maa and Mehta,
1987).
Table 6. Bed recovery in Test No. 9 of Feng (1992)
Elevation a'(Pa) Recovery
(cm) t=0 t=7.5 hr t=11.5 hr [(0 1.5r7.5)/(a0o07.)]xl00
12.0 39 8 8 0
9.5 53 0 41 77
7.5 67 7 35 47
5.1 83 53 59 20
presumably reflects local variability in the mud properties at the sites of pressure measurements. In general,
however, with the exception of the data point at 12 cm (4 cm below the interface), recovery is observed to slow
down with increasing depth. Overall, since the attapulgite + kaolinite mixture was thixotropic, the rate of recovery
is likely to be related in some way to the rate of gelling. Thus, for example, Day and Ripple (1966) reported a
recovery period of 14 hr for an initially sheared Kmontmorillonite based on the measurement of suction in a
tensiometer. Typically, gelling is complete in about a day.
ENERGY DISSIPATION AND ELEMENTS OF MODELING
4.1 WAVE ENERGY DISSIPATION OVER NONRIGID BEDS
The conservation equation for wave energy within the water column is:
a E (7)
a+ Vh'CgE)+ = 0 (7)
where E = mean wave energy per unit surface area, Vh = horizontal gradient operator, Cg = wave group velocity
and eD = rate of dissipation of wavemean energy per unit area. Constitutive models proposed for the rate of energy
dissipation within mud generally fall into two groups: 1) models which are specific to the assumed description of
mud rheology, and, 2) models which are nonspecific with regard to rheology. Some of the proposed expressions
for ED are listed in Table 7. In the first group, cohesive mud has been considered either as a viscous fluid, or as
a viscoelastic or poroelastic medium. Some investigators, e.g. Shibayama et al. (1990), have also considered
cohesive mud to be a Bingham viscoplastic in which, at stresses less than the yield value, the material is treated as
a viscoelastic, whereas at higher stresses it is a fluid. Silty muds, considered to be poroelastic, incorporate energy
loss by Coulomb friction between clay particles (e.g. Yamamoto and Takahashi, 1985).
In the second group, the expression of Tubman and Suhayda (1976) requires coefficients M and q which
depend on the character of the water pressure wave near the mud surface and that of the induced mud surface wave.
Since these waves depend on the solution of the hydrodynamic problem with an assumed constitutive relationship
for mud, M and 0 must be measured directly, or obtained indirectly via model calibration using experimental data.
Likewise, the expression of Schreuder et al. (1986) involves the mean shear strain rate as an explicit input
parameter. Appendix A provides additional information on the bases of the ED expressions.
4.2 A MODELING PERSPECTIVE
The response elements shown in Fig. 1, namely wave attenuation, mud motion and surface erosion, do not
easily lend themselves to a unified theoretical treatment, since the first two can be examined, at least in the
simplified treatment, by assuming bottom mud to be a constant density, incompressible continuum, whereas
particulate entrainment at the mudwater interface requires separate consideration in terms of the change in the
density of the overlying water layer due to a fluid stressdependent vertical mass flux at the interface. Even under
the most severe waveinduced agitation of bottom mud, the lack of a competent diffusive mechanism to transport
the sediment mass from the bottom to the top of the water column usually precludes the development of high
suspension concentrations in the surficial waters; most of the material remains close to the bed, approximately within
the wave boundary layer. In turn, the depth of scour of bottom mud is typically sufficiently small to make it
practicable, at least to first order accuracy, to assume water and mud thicknesses to be timeindependent. Under
Table 7. Some energy dissipation rate expressions due to nonrigid bed mechanisms
Mud property" Expressionb Investigator(s)
Group 1
Mud as viscous
fluid1
Mud as viscous
fluid2
Mud as viscous
fluid3
Mud as Voigt
body4
Coulomb friction
in bed5
Coulomb friction
in bed6
1/2
1 f12 222 2krh )2 2 2
e ( w2gkwr) a
CDI + CD2 + D3
CD4 + CD5
1 i 2 2(gkcoshkhi 02sinhkhi)
pig >A2Im
2 r gk(gksinhkh1 w2coshkh1)
PS2 w)a2
4 Gkocosh2koh
4 Gkocosh2lkohl (1W/) 2*2
Dean and Dalrymple (1984)
Dalrymple and Liu (1978)
Feng (1992)
MacPherson (1980)
Yamamoto and Takahashi
(1985)
Yamamoto and Takahashi
(1985)
Pig M sin4a2
2 cosh2krhJ
i2 ('<>2h
Tubman and Suhayda (1976)
Schreuder et al. (1986)
aSee Appendix A for explanatory notes 1 through 8.
bNotation: 1 = water layer; 2 = mud layer; p = density; v = kinematic viscosity; w = wave frequency; k =
complex wave number = kr+iki, where k, = wave attenuation coefficient, kr = wave number and i=(1)12; h =
depth; g = acceleration due to gravity; a = local wave amplitude; a0 = reference wave amplitude; CD1 through
eD5 = rate of energy dissipation terms; Ar = wave amplitude ratio; Im = imaginary part of a complex variable;
S= Coulomb specific loss; G = dynamic shear modulus; ko = wave number for a rigid bottom; w02 = 2Gk02/p2;
M = proportionality constant between the amplitudes of the mud wave and the waveinduced bottom pressure wave
; 4= 18000, where 0 = phase angle between the crest of the bottom pressure wave and the crest of the mud wave;
pu' = dynamic viscosity;
= average value of shear strain rate.
Group 2
Mud wave7
Mud as
viscoelastic
medium8
these assumptions, solutions developed for problems involving stratified, nonNewtonian fluid mechanics have been
applied extensively to model wavemud interaction. Some of the previous developments are summarized in
Appendix B.
In what follows, the formulation of the linearized hydrodynamic problem for wave attenuation and mud motion
is considered in conjunction with Newtonian viscous and linear viscoelastic descriptions of mud rheology.
Progressive, nonbreaking waves are treated for their propagation within shallow water and nonshallow
(intermediate depth) water conditions. Following this development, simple bases for modeling erosion are discussed
briefly. These presentations are for informative purposes only; for details the referenced works in Appendix B must
be consulted.
4.3 BASIC CONSERVATION EQUATIONS
For continuous media, the governing equation of continuity is:
Dp + pui 0 (8)
Dt Oax
where p(x1,t) = density, ui(xi,t) = velocity and subscript i denotes direction. For an incompressible fluid Dp/Dt=0,
and the continuity equation is simplified to
aui = 0 (9)
axi
The corresponding equation of momentum is:
Dui T+ i (10)
p = pb. i (10)J
Dt ax.
where bi(xi) = body force, Tji(xi,t) = external surface stresses, and subscripts i and j denote directions in
conjunction with subscript ji, which implies a second order tensor. Thus, j = direction normal to the surface and
i = stress direction. Because gravity is the only body force of concern here and as we set the zdirection vertically
upward, then bl=b2=0 and b3=g. The stress, Tji, can be expressed as the sum of the deviatoric component,Tj,
and the spherical component, 1Tkkai=PSi, where Tk is the sum of the diagonal terms in the stress tensor, p is
the mean normal pressure and 6j = kronecker delta; forj=i, 5ji=l, and for ji, fji=0. Thus,
Du. ap +Tj (11)
p = P3ib3 xj
Dt oaxi axj
4.4 CONSTITUTIVE EQUATIONS
4.4.1 Fluid incompressiblee)
The constitutive equation for the fluid is:
Tj = 2uD: (12)
where t = viscosity, and the rate of strain, Dj =0.5[(auj/axi)+(au,/axj)]. Thus, the momentum equation becomes
Dui ap i2ui (13)
P = pIb3 i Ax axj
Dt Ox. axax1
4.4.2 Elastic Material
The constitutive equation for elasticity is:
Ti = 2GEi (14)
where G = modulus of elasticity, and the strain Eji =0.5[(aLj/axi)+(aLi/axj)], where L, Lj are displacements.
Thus the momentum equation becomes
Dui D 2Li ap ( L
p = P i p3ibi +i + G_ (15)
Dt Dt2 i x a ax
4.4.3 Viscoelastic Material
The general constitutive equation for a linear viscoelastic medium is
M ar N a' (16)
EPr CTji E, ) (16)
rO at' s0 at'
where subscripts r and s denote orders of partial differentiation, pr and q, are coefficients related to the viscoelastic
model, and M and N are the specified maximum differential orders of the chosen model. Under cyclic loading, we
consider
Tj = Toexp[i(wt)] (17)
Ej = E0exp[i(wt8)] (18)
where ( = angular frequency, 5 = phase shift, TO and E, are amplitudes of Tji and Eji, respectively, and i=(
1)1/2. Therefore, the relation between the deviatoric stress and the strain becomes
N
q,(i")s
Tji 0 (19)
#' M
Eji Pr,)r
rO
and the corresponding relation between the deviatoric stress and the rate of strain is:
N
E q,(iw)'
Tji 0 (20)
M
Eji Pr(w)r
rO
After some manipulation the result is
... (21)
Tji (&' +ij")Eji =(Eji
where the dot denotes timederivative, i*=p'+ip" is the complex dynamic viscosity, p' = real part, or the
dynamic viscosity, and p" = imaginary part, or second viscosity. Thus the momentum equation becomes
SDu + (p'+ij") a2ui (22)
Dt ax axj axj
For a fluid:
/ 2' = 0 ; V" 0 (23)
For an elastic material:
p' = 0 ; G" =G (24)
Ci
For a viscoelastic material Voigt (Kelvin) model:
A (25)
Cc' f ; /z" ffi 
Note that for a viscoelastic medium, from Eq. 19 we can also define Tji =G*Eji, where G*=G'iG", where G'
is the elastic energy storage modulus and G" is the viscous energy dissipation modulus. Thus it can be shown that
for the Voigt model, for example, G'=G and G"=u2w (Barnes et al., 1989).
For a viscoelastic material Maxwell model:
S= 212 ; =" = 2Gw (26)
2+ G 2+ G
A2 A2
Next, as illustrations of more complex models, for the threeelement viscoelastic models, constructed from Voigt
and Maxwell elements, shown in Figs. 18a,b:
Model (a):
G, 2G (G,+G2)G2
2G, 21 2
2 ; =2 (27)
Gg + Gl+GZ
S+G 2 G+
This model has been used in describing the consolidation behavior of soils (Keedwell, 1984). Jiang (1993) used it
in his wavemud interaction model described in Section 4.5.
Model (b):
2G2 w 2
2 P2 (28)
G [I 2
ffI r + ; G 2=1 +W
A2 A2
For any other (linear) viscoelastic model, the complex dynamic viscosity, *=/' +ip", can be obtained through
Eq. 20.
4.5 MODELS FOR WATER WAVES OVER MUD (XZ PLANE)
4.5.1 Inviscid, Shallow Water over Mud
A general definition diagram for the wavemud system is shown in Fig. 19. In the case considered first, water
is assumed to be inviscid and mud to be viscoelastic. Under the shallow water assumption, the following analytical
solution of the linearized problem is obtained based on the works of Gade (1957, 1958) and Jiang (1993).
For water:
The continuity Eq. 9 is integrated from the watermud interface to the water surface:
Sh+u d z (29)
h+fz x az
2G1
2G1
22] 2 G2 2G2
(a) (b)
Figure 18. Threeelement viscoelastic models: a) Voigt element (represented by 2/C2, 2G2) modulated by additional
elasticity; b) Maxwell element (represented by 22, 2G2) constrained by elastic modulus represented by 2G1.
A
v "' Wl(x,z,t)
Water hi u u1 (x,z,t)
w2 (x,z,t)
Mud h2 _U2 (x,z.t)
/]//////////////// ,]/ ].
Figure 19. Watermud system in the xz plane.
in which water velocity, u1, is invariant over water depth, and at the surface and the interface:
a ar
wlZ'^ =  ; 'wlz.hir, = a
Therefore, the continuity equation for water becomes:
hau1t (+ 2)
Sax at
From Eq. 13, omitting the viscous terms, /a2ui /a xj axj, for inviscid "water", using the hydrostatic assumption
for pressure, p=plg(~1z), in shallow water and omitting the nonlinear terms, u Ou1/axj, the momentum equation
for water (in the xdirection) becomes
au ap 81 (32)
1t ax ax
For mud:
Equation 9 is integrated from the mud bottom to the watermud interface using:
wlhh2= 0 ; wIh+. (3)
Then the continuity equation for mud becomes
h dz + 0 (34)
Ox at
(h +h2)
From the momentum Eq. 22, using the hydrostatic pressure assumption and omitting the nonlinear terms, we
obtain:
Pu22 a8l 2 g + 2u2 (35)
P = pig (p2P1)g
at Ox Ox az2
where P2 = mud density. The horizontal diffusion term, U*(a2u2/ax2), has been ignored in comparison with the last
term on the right hand side (in Eq. 35) corresponding to vertical diffusion. Eq. 35, in general, is applicable to small
strain oscillatory forcing.
The following boundary conditions are next introduced:
At the free surface, z=0:
rl(0,t) = Acos(wt) (36)
where A = wave amplitude.
At watermud interface, z=hl:
au2(x,hlt) 0 (37)
az
At the mud bottom, z=(h1 +h2):
u2[x, (hI +h2),t] =0 (38)
Under cyclic wave loading, the following harmonic assumptions occur:
S= fl exp[i(kxwt)]
2= '2 exp[i(kxwt)] (40)
ul =ai exp[i(kxwt)] (41)
u2 =k2(z) exp[i(kxwt)] (42)
where the amplitudes denoted by ^ are independent of x, z and t except in Eq. 42, wherein 02 varies with z. Next,
substituting the boundary conditions into the governing equations (31, 32, 34, 35) and after some manipulation the
following solution for the complex wave number, k, is obtained:
F1 2 11 112
1+ r 1+ r 4rhr
F, h h, i h (43)
k2r r
hi
where r=(p2p)/p2; Froude number, Fr=w(hl/g)112; r= [tanh(mh2/hl)/(mh2/hi)]; m=(Re/i)1/2 and Reynolds
number, Re= whl2/v (v =* /P2). Note that k=kr+iki, where kr is the wave number and ki is the wave attenuation
coefficient. Thus the wave amplitude at a distance x, ax=aoexp(kjx), where ao is the amplitude at x=0. Solutions
for the other variables are:
'1 = Aexp[i(kxwt)] (44)
kh1 (45)
kh,
S fA 1 1r exp[i~kxwt)]
u = A exp[i(kxwt)] (46)
Fr
u = wA 1cosh m +tanh m sinh m (h1
F2 h h h,
Fr
(47)
[1r 2 exp[i(kxwt)]
Finally, the shear stress in the water layer, hence at the interface, is zero because of the inviscid assumption,
and in mud layer the shear stress is obtained from:
aug
T(x,z,t) = [ 
dz
[ m z+(hl+h2) 1 m h 1 z+(h,+h2) 4
= sinh m_ + tanh m cosh m (48)
h1i T hJ h4
kh khni 21 at
*A 1r exp[i(kxot)]
Fr I Fri
Without invoking the shallow water assumption, Hsiao and Shemdin (1980) and MacPherson (1980)
independently investigated the problem of inviscid water waves propagating over a linear viscoelastic mud, and
obtained analytical solutions in which the wave number is implicitly expressed. Suhayda (1986) extended
MacPherson's results to a nonlinear viscoelastic bed, while Mei and Liu (1987) considered a viscoplastic bed. Foda
(1989) examined a stratified elastic bed and Feng (1992) and Isobe et al. (1992) set up their models considering mud
to be a nonNewtonian (powerlaw) fluid. However, as noted, since in these models the viscosity of water was not
included, the shear stress on watermud interface, which governs particulate entrainment at the interface, was
neglected.
4.5.2 Viscid Fluid (Water) over Viscid/Viscoelastic Mud
Considering the viscosity of water and nonshallow water conditions, the governing equations can be set up
as follows:
From Eq. 9 the continuity equation is:
8ui awi (49)
Ox az
where u and w are velocities in the x and zdirections, respectively, and i= 1,2 denote the upper (water) and lower
(mud) layers.
From Eq. 22 for a viscid fluid, omitting the nonlinear terms, the momentum equation components are:
ui 1 api + i 2 a2ui (50)
at pi p x a x az2
Owi 1 api P Ia2 a2wi (51)
at p 9z a ax2 z2
where the dynamic pressure, Pi, is defined as:
Pi P! + Pigz + P? (52)
Here, Pit = total pressure, and
O when i=1 (53)
PO
S (p2Pl)gh1, when i=2
The viscosity, vi, depends on the medium; for water, vi=vl, while for mud, vi=v 2=;*/p2 from Eq. 21. The
complex viscosity, p*, depends on the chosen constitutive model for mud rheology; see, e.g. Eqs. 23 through 28.
The following ten boundary conditions are introduced next:
At the mud bottom, z=(h +h2)
w2 = 0 (54)
U2 = 0 (55)
At the mudwater interface, z=h +2
w2 = wl (56)
u2 = u (57)
Sr2 (58)
W2 
Pt awl a w2 (59)
vI"2pjv a Plgr2 Pi o2"22T P2gr2
au, aw, a 8u aw2 (60)
1l + 2 
9z ax az ax
At the free surface, z= rl
= (61)
at
p t 2p,;,awl 0 (62)
lPl0 .ul + awl 0 (63)
l z ax
Under cyclic wave loading, consider:
r = i(z)exp[i(kxwt)] (64)
ui= i(z)exp[i(kxwt)] (65)
wi = *i(z)exp[i(kxwt)] (66)
Pi = Pi(z)exp[i(kxwt)] (67)
and for the shear stress:
aui awi 1
Sa (68)
= p a'i(z +'*i(z) ik exp[i(kxwt)]
Implicit solutions for the wave number and attenuation coefficient, velocity, pressure and shear stress are
obtained (Dalrymple and Liu, 1978). Maa and Mehta (1987,1990), Chou (1989) and others extended this method
to incorporate stratified mud properties. Jiang (1993) has extended the approach to include second order effects
arising from finite amplitude waves.
4.6 MODELING WATER COLUMN SEDIMENT DYNAMICS
Since we are focussing on vertical transport in the water column, the basic equation is the balance of sediment
mass in terms of the settling and upward mass diffusion fluxes, F, and Fd, respectively:
8PD + (dF,)=0 (69)
at Dz
where PD = OPs = (PPw)Ps(PsPw) is the dry density or mass concentration of sediment in suspension (dry sediment
mass per unit volume of suspension). We note that Fd=ksPD/aZ, where k, is the vertical mass diffusivity, and
Fs=WsPD, where w, is the settling velocity. Given the comparatively short wave periods in relation to the gradually
varied trend in pD, it is customary to consider vertical transport on a waveaverage basis.
The initial condition for solving Eq. 69 is prescribed by pD(z,0), and at the water surface, z=0, the usual no
net flux condition is, Fd(0,t)F,(0,t)= 0. Two approaches have been used in prescribing the erosion flux condition
at the mud surface, z=hi, as follows.
Some investigators, e.g. Thimakorn (1980; 1984) and Cervantes (1987), have prescribed pD(hl,t), following
a similar approach commonly used in coarsegrained sediment transport. In that case a reference concentration is
prescribed close to the bed at elevation z=hI + 5a, where 5b is a small distance above the bed which depends on
the work done by the fluid in dislodging and raising the potential energy of the surface particle elevated to 6a
(Smith, 1977). Since, however, bed load transport does not occur when the sediment is finegrained, especially
cohesive (Partheniades, 1977), for most practical purposes it is reasonable to set 5a=0. In any event, defining
((t)=PD(hl,t)/PD(t), where PD(t) is the instantaneous depthmean value of pD(t), experimental data for 3(t) and
an empiricalfit equation are given in Fig. 20. The tests (Tl, T2 and T3) were performed in a waveflume shown
schematically in Fig. 21, using an estuarine mud. See Section 5.2 for a further description of the flume. The wave
frequency &, and the nominal wave amplitude, ao, for each test are given in Table 8. Other relevant test parameters
are prescribed in Cervantes (1987). In the empirical equation of Fig. 20, =0/00Io, 34=,4/061 and i =8it; thus data
fit for P(t) is dependent on 0,, 94 and 61. These values are given in Table 8.
NORMALIZED TIME, t
Figure 20. Variation of with normalized time, t (after Cervantes, 1987).
Roughened Beach
^ x No
  Water  
Wave Maker
Figure 21. Wavemud interaction flume (after Maa, 1986).
Table 8. Test parameters including wave frequency w and nominal wave amplitude, a0, and coefficients Po0, 4 and
61 for the empiricalfit equation in Fig. 20
Test w ao 0o 94 a1
(rad/s) (m) (s1) (s1)
T1 6.3 0.030 18.5 0.045 0.00055
T2 3.1 0.015 50.0 0.220 0.00090
T3 6.3 0.035 48.0 0.060 0.00050
A noteworthy observation relative to Fig. 20 is that j(t), hence 3(t), is observed to rise rapidly initially,
followed by its approach to a steady state value close to one. The initial rise represents a mass erosiontype bed
failure phenomenon represented by the MohrCoulomb equation; see Lamb and Whitman (1969), whose effect is
mainly embodied in the coefficient P4, which, for a given bed, apparently depends not only on rb, but also on
8rb/8t (Cervantes, 1987). In Fig. 20, the trend of increasing mass erosion with increasing P4 is generally evident.
This characteristic response of the bed suggests that the release of fine sediment into the water column from the bed
is likely to be an episodic process, with masseroded bottom sediment entrained perhaps by coherent bursts. The
approach of J to unity suggests that following the timedependent effects associated with experimental startup, the
vertical dry density profile became uniform. This was so because pD(hl,t) was actually measured above the
lutocline. Thus, fluid mud generation is not represented in the character of &(t) in Fig. 20, for which the datum,
z=hl, was located approximately at the elevation of the lutocline. Thus, in effect 5, was approximately equal to
the thickness of the fluid mud layer.
It can be easily shown (Cervantes, 1987) that by specifying 3(t), and introducing it in Eq. 69 renders the
problem as one of initial value, which can be solved for pD(t) given PD(0), and it is recognized that PD(t)
approaches a constant value, PDS, as too. The vertical distribution of pD is not obtained by this method.
In low energy environments where significant fluid mud generation usually does not occur, = 1 is not an
unreasonable assumption. Elsewhere, however, in general wavegenerated fluid mud must be treated explicitly as
an eroded sediment mass, especially since in this state mud is relatively easily transported horizontally by currents.
Process models for fluid mud generation have been developed (e.g. Chou, 1989; Foda et al., 1993) to calculate the
fluid mud thickness (see Section 3.2).
Erosion of the fluid mudwater interface occurs as a process involving shearinduced warping of the interface,
which leads to its destabilization and mixing (Scarlatos and Mehta, 1993; Adams et al., 1993). This process is
mainly dependent on the Richardson number and also the settling velocity of the particles being entrained.
Experiments under unidirectional shear flows have yielded the following expression:
PlAbU A B (70)
S + DRi
g Ri Pe 1/2
where pi = mean upper (mixed) layer density, b = buoyancy = g(p2Pw)/Pw, P2 = fluid mud density, pw = water
density, Ab = buoyancy jump across the fluid mudwater interface, Ri = Richardson number = hlAb/U2, U =
mean velocity of the upper layer, Pe = Peclet number = Uhl/km, km = molecular diffusivity and A,B,D are
empirical constants (Mehta and Srinivas, 1993). The coefficient D depends on the settling velocity; thus, a non
settling constituent such as salt D=0, and Eq. 70 becomes applicable for example to salt entrainment. Note that the
Peclet number dependent term is important only at very high values of Ri when the flow becomes viscous. For most
practical applications it is appropriate to set B=0. The applicability of Eq. 70 under a wave field remains to be
demonstrated, however.
The second approach with respect to the erosion flux involves the specification of apD(hl,t)/8t (Ross and
Mehta, 1989). Thus, from Table 5,
aPD(hl,t) M2 b (71)
at h, Ir(t)
in which 7R(t) is obtained from rR(z) via the continuitybased zt transformation
az = h PD(t) (72)
Tt p2(z) at
where p2(z) is the depthvarying bottom mud density. In the difference form, Eq. 72 can be written as
At=Az/{[hl/p2(z)][apD (t)/Ot]}. Thus, a change in 7R over the differential scour depth, Az, can be interpreted in
terms of the corresponding change in TR over At. Note that, since TR typically increases with p, by virtue of Eq. 71
scour is arrested when the condition, R= Tb, is attained at a particular depth.
The datum, z=hI, can be conveniently located at the level between the bed and fluid mud. In this way fluid
mud can be treated in an approximate way as a suspension formed by bed erosion specified by Eq. 71. This equation
must be "calibrated" by adjusting the value of M2 to account for the rate of generation of fluid mud, as well as its
entrainment into the water column above the lutocline.
In high energy environments wherein a considerable amount of sediment is resuspended, the influence of the
negative buoyancy and turbulence damping characteristic of the sedimentwater mixture can be substantial, and tend
to retard upward mass diffusion (Wolanski et al., 1988). It is therefore essential to amend the commonly chosen
neutral diffusivity, ko, by the damping function 4= 1 +az/L, where a5 is an empirical coefficient, and L is the
MoninObukov length scale. Thus, ks=ko4, and a common empirical form for 4' is: =(1 +f6Rig) a, where the
gradient Richardson number, Rig= (g/p)(dp/dz)/(du/dz)2, p is the density of the sedimentwater mixture and u(z)
is the horizontal flow velocity. We therefore have the diffusion flux, Fd=ko(l +36Rig) aP D/aZ, which is plotted
in Fig. 22 against 8pDlaZ for stable and unstable stratifications, given Cg6=2, 6=4.17 and other relevant parameters
based on the work of Ross (1988). See, e.g. Horikawa (1978) for a discussion on the choice of ko and Ross (1988)
for cr6 and 16 values.
The settling velocity, w,, in general varies with pD. Defining PD1 such that for all PD
i.e. the interparticle collision frequency is so low that particles or aggregates settle essentially independently of each
other, the following expressions can be used:
For PD
ws = constant (i.e. independent of PD) (73a)
Figure 22. Diffusion flux, Fd, as a function of apD/aZ for ac6=2 and 06=4.17 (modified from Ross, 1988).
For pD>PD1,
n
w apD (73b)
w (p+b2)"
where coefficients a, b, m and n depend on sediment properties. In addition, as noted later, they also depend on
the degree of agitation in the water column. In Eq. 73b, for relatively low values of pD, b2> >pD2, hence w,=ab"
2mPDn. This is called flocculation settling in which w, increases with PD due to a corresponding increase in the inter
particle collision frequency. At high values of PD, b2<
(m> n/2) with increasing PD due to hindered settling.
Settling velocity data that approximately conform with Eq. 73b are shown in Fig. 23 with a=33.38, b=2.537,
n=1.83 and m=1.89. The settling flux, Fs, is obtained from
n+l
F aPD (4.68)
G(p+b2)m
The quantity pDI is somewhat arbitrarily prescribed in Fig. 23 as 0.1 g/l, pD2 = 2.46 g/l is the value of pD at
which the settling velocity peak occurs, and pD3 = 4.38 g/l corresponds to the peak value of F,. A lutocline occurs
when PD exceeds pD3[=(pPuPw)/(Pslw)].
For a given sediment, the coefficients a, b, m and n are found to be sensitive to the degree of agitation, or
turbulence, in water. As observed from Fig. 24, agitation by vertically oscillating rings in a laboratory column
caused the particulate aggregates to rupture and consequently settle more slowly than under quiescent conditions
(Wolanski et al., 1992).
APDn
1 Ws= (PD2 + b2)
101 *
PD1 = 0.1 g/
PD2 = 2.46 g/I
PD3 = 4.38 g/I
4n31 I_ I I
U)
 100
101
C
X
102 n
103 2
102
Figure 23. Settling velocity (and flux) variation with pD. Data points obtained from tests using a lake (Okeechobee,
Florida) sediment in a laboratory column (after Hwang, 1989).
>" 10
UJ
10"1
z
w19
Iu101 1 10
DRY DENSITY, PD (g/l)
Figure 24. Settling velocity versus PD data using a bay (Townsville, Australia) sediment under two conditions:
quiescent water and water agitated by vertically oscillating rings (adapted from Wolanski et al., 1992).
102 
S102
101 1 10
DRY DENSITY, PD (g/l)
Maximum Ring Velocity 0.9 cm/s

L
LABORATORY/FIELD DATA AND MODEL SIMULATIONS
5.1 INTRODUCTION
As noted, given the compliant and highly dissipative nature of bottom mud, waveinduced water motion is
significantly contingent upon the choice of the constitutive relationship describing mud rheology. For illustrative
purposes we will consider mud to be viscoelastic, a special case of which is mud as a viscous fluid. Note that mud
motion under typical conditions, both in the laboratory and in the field, is sufficiently damped by the high viscosity
of mud, which tends to be order of 103 to 104 times greater than water. Thus, turbulence within mud can be ignored
in most practical applications, even though it does occur to some degree below the lutocline, and contributes towards
the entrainment of fluid mud (Wolanski et al., 1988). Laboratory applications are described first, followed by field.
5.2 WAVE FLUME RESULTS
5.2.1 Test Facility, Experiments and Modeling Approaches
Experiments were conducted in a 20 m long, 46 cm wide and 45 cm high plexiglass flume shown in Fig. 21.
Progressive waves were generated by a programmable, plungingtype generator. Mud was contained in a 16 cm
deep, 8 m long trench with end slopes as shown. The downwave end of the flume beach was artificially roughened
to absorb wave energy and thus minimize reflection. Details of flume description are found in Ross (1988). Muds
of different composition, degree of consolidation and vertical density structure were tested. Figure 25a shows the
timevariation of stratified density profiles of a consolidating mud from Cedar Key, Florida (designated CK)
prepared by depositing the sediment from an initially formed suspension in the flume. Note that, in general, in such
experiments the timescale of measurable density variation is considerably greater, on the order of tens of minutes,
than the typical wave period and, therefore, consolidation does not measurably contaminate the waveaveraging
process usually inherent in modeling. In contrast to Fig. 25a, Fig. 25b, shows a practically uniform clayey mixture
composed of an attapulgite and a kaolinite (designated AK) in equal proportions by weight. The bed was prepared
by pouring a premixed sediment slurry over the flume bottom.
All experiments were conducted with a nominally constant mud thickness, a constant water depth, and
monochromatic, progressive waves of selected frequency and forcing amplitude. Measurements in general included
density profiles from the water surface to the flume bottom, wave height decay over the length of the trench, mud
elevations along the trench, water velocity, mud velocity and acceleration, and vertical total and pore pressure
profiles. For the types of transducers and measurement techniques used see Maa (1986), Ross (1988) and Jiang
(1993).
The presented experimental results encompass three categories: wave attenuation, water and mud dynamics and
bed/interface erosion, in consonance with Fig. 1. With respect to wave attenuation and mud dynamics the data are
compared with simulations using the models of Maa (1986) and Jiang (1993). Both consider water to be Newtonian
and mud to be a viscoelastic continuum. The equations for continuity and the horizontal and vertical components
of motion for these models are given by Eqs. 49, 50 and 51, respectively. The basic problem formulation for both
MUD DENSITY (kg/m3)
Q *" Run1'
g 12
o,2
0.05
0 4 0.05
LU01t
0 0.05
0.10
0.10
nn.i .
(b)
F

Flume Bottom
,
0.15 0.20
1100 1200 1300
MUD DENSITY (kg/m3)
1400
Figure 25. a) Timevariation of the bottom mud density profile during an erosion experiment (Run 1) using
mud from Cedar Key, Florida (after Maa, 1986); b) Uniform density profile of a mixture of an attapulgite and a
kaolinite (after Jiang, 1993).
r
, 4.0
OE
m 8.0
zo
C 12.0
wJ
tu 16.0
)
20.0 I
10(
00
models is given in Section 4.5.2. Differences between the two models are as follows: The small amplitude solution
of Maa (1986) is based on linearized boundary conditions. In Jiang (1993), the nonlinear, finite amplitude solution
is carried out to second order using the perturbation expansion technique. In Maa (1986), bottom mud can be sub
divided into as many layers as desired, each specified by the density and the two viscoelastic constants for the Voigt
constitutive model (Eq. 25). Thus stratified mud, such as shown in Fig. 25a, can be parameterized in this model.
In Jiang (1993) the bed properties are vertically uniform, such as represented by the density profile in Fig. 25b,
and the viscoelastic model is given by Eq. 27, which is more general, and approaches the Voigt model at
frequencies greater than about 10 Hz. Neither model includes interfacial erosion; hence the bottom mud mass is
conserved.
The viscoelastic assumption causes the kinematic viscosity, P2, in the equations of motion (Eqs. 50 and 51) for
mud to be a complex number, /*/p2, the form of which depends on the choice of the constitutive model, as noted
earlier. In general, for a given mud and temperature, j* varies with the forcing amplitude, frequency and mud
density (Jiang, 1993). As noted in Section 3.2, in a controlledstrain rheometer, Chou (1989) found the strain
amplitude to be an important parameter in determining mud response (viscous, viscoelastic or elastic), and frequency
to be of secondary importance within the range of frequencies examined. In a controlledstress rheometer Jiang
(1993) examined mud response to oscillatory stress amplitude as well as forcing frequency. Several muds were
tested, the stress amplitude was varied from 0.3 to 25 Pa, and frequency from 0.02 to 4.5 Hz. Typically, for stress
amplitudes ranging from 0.6 to 10 Pa, the response was found to be relatively weakly dependent on the amplitude.
On the other hand, frequency dependence was more significant. As an example, in Fig. 26, the magnitude of the
equivalent viscosity, I * = Ip2v*I =p2(L2+tr"2)1/2, is plotted against frequency, f, for a mud from offshore of
Mobile Bay, Alabama (designated MB), the site of a dredged material deposit from the bay (Mehta and Jiang,
1993). The strong dependence of Ij* I on f is believed to be due to the thermodynamic process associated with the
vibratory response of mud to cyclic forcing; at low frequencies the process is probably isothermal, while at high
frequencies it tends to become adiabatic (Krizek, 1971; Schreuder et al., 1986). This change evidently has the
overall effect of decreasing j *1 with increasing f. Increasing 0 in turn increases the stiffness and the effective
viscosity.
The effective viscosity magnitude in Fig. 26 has been determined for the viscoelastic model shown in Fig. 18a,
and is thus defined by p, G1 and G2. These three parameters were experimentally found to depend on f according
to:
IA, G1 or G2 = exp(). fA (75)
where values of the coefficients e and A are given in Table 9 (Mehta and Jiang, 1993). Relationships such as the
ones in Fig. 26 inherently suggest that, for example, as waves pass over mudbanks the wave spectral shape may
be drastically altered due to frequencyselective attenuation, as observed for example off Surinam in Fig. 27 (Wells
and Kemp, 1986).
101 100
FREQUENCY, f (Hz)
Figure 26. Magnitude of equivalent dynamic viscosity (applicable to the viscoelastic model of Fig. 18a) as a
function of wave forcing frequency and mud volume fraction, 0, for MB mud.
Table 9. Coefficients of Eq. 5.1 for MB mud
0 G1(Pa) G2(Pa) p(Pas)
e A A e A
0.07 3.659 0.030 1.439 0.975 3.165 0.975
0.11 6.352 0.075 2.139 0.745 6.695 0.745
0.17 8.274 0.108 3.864 0.696 8.374 0.696
Waveinduced erosion is separately examined in Section 5.2.3 via the approaches described in Section 4.6,
following wave attenuation and associated mud dynamics. The first approach yields the timevariation of the depth
averaged suspended sediment concentration or dry density, pD(t), during erosion, while the second gives the time
variation of the vertical profile of concentration, pD(z,t). Some relevant laboratory results are considered next.
5.2.2 Wave Attenuation and Mud Motion
In Fig. 28 the wave attenuation coefficient, ki=(l/x)ln(ax/ao), measured over a distance approximately equal
to the length of the mud trench (Fig. 21), has been plotted against wave frequency, w, for AK mud. The depth of
Figure 27. Wave spectra from synchronous wave stations 22, 11 and 4 km offshore, showing loss of wave energy
and change in the shape of the spectrum over a fluidlike mud bottom along the central Surinam coast. HIs = root
mean square wave height, T = wave period and h = water depth (after Wells and Kemp, 1986).
water (hj) in these tests was 16 cm, while the mud depth (h2) was varied. The input wave amplitude (=ao) was
1.3 cm. In addition to the effect of water depth on the degree of attenuation, the maximum in the kw curve is an
interesting and characteristic feature that requires an explanation. In the special case of shallow water and viscous
mud (see Section 4.5), Jiang and Mehta (1992) showed that the degree of attenuation, as reflected by kl, depends
on the ratio, h2/1m, where 6m=(2v2/w)1/2 represents the viscous mud boundary layer thickness. The depthmean
value of the velocity gradient, au/az, and in turn the rate of wave energy dissipation in mud, is maximum when
h2/bm is equal to or is close to one, and decreases as h2//m increases above one or decreases below one. This
criterion is modulated when mud rigidity is measurable. Thus, MacPherson (1980) showed that the peak wave
52
LL
UJ
III
Op'
Z
L0
<
0.4
0.2
0.0 I I I I
0 3 6 9 12 1
ANGULAR FREQUENCY, (o (rad/s)
Figure 28. Wave attenuation coefficient against frequency for two AK mud (4 = 0.16) depths, 12 cm and 18
cm. Water depth was 16 cm. The input wave amplitude was 1.3 cm. Circles are data; lines are simulations
based on the model of Jiang (1993).
n
z
0
F 0.2
W 0.4
H
W 0.6
5 0.8
cc
w
_1
n___
rI 
0.00
0.00
0.05
0.10
0.15
0.20
0.25
ACCELERATION AMPLITUDE (m/s2)
Figure 29. Comparison between measured and modelcalculated profiles of the amplitude of horizontal acceleration
in a watermud (AK) system. The water depth was 16 cm, mud thickness 18 cm, water surface forcing wave
amplitude 0.5 cm, wave frequency 1 Hz and 0 = 0.12 (after Jiang, 1993).
I Id I
AK Mud Simulation
h2 = 18 cm
Water
Interface Calculated
J
Mud *
Data
AK Mud
I II II
.
I
n
attenuation rate occurs when I = v2/(gh)12 and G =G/p2gh1 are of the same order of magnitude. In any event
it is evident and noteworthy that, as a result of the strong dependence of ki on w, the mode of change in the wave
spectral shape as the waves pass over a mudbank will critically depend on the character of the input spectrum, mud
viscosity and mud depth.
Based on the results from another test using the same mud (AK), a comparison between the measured and
modelcalculated amplitude of the horizontal acceleration, 42, within mud is shown in Fig. 29. Note the drastic
reduction in mud acceleration relative to that in the water column due to the dissipative nature of the bottom
medium.
A noteworthy feature of bottom mud subjected to nonlinear waves is residual mud mass transport, which in
a large number of situations is a significant factor leading to sedimentation, e.g. in port channels and basins. The
Lagrangian mean particle drift, or Stokes' drift, lL, associated with "nonclosed" trajectories of mud particles must
therefore be calculated. In general, the velocity of a particular particle with a mean position of (xl,zl) is u(x1 + r',
z + '), where r' and t' are instantaneous position coordinates along the particle trajectory. Based on Taylor series
expansion, we have (Sakakiyama and Bijker, 1989; Jiang, 1993):
UL(Xl+r1+') u 1) + Ju2(xl,zl)dt + Jw2(xzi)dt
where overbars denote timeaveraging over the wave period and i2 is the Eulerian mass transport velocity. To obtain
uL theoretically, the waveaveraged intergals and derivatives on the right hand side of Eq. 76 must be obtained
through an appropriate hydrodynamic model. In Fig. 30a comparison is made between measurement and prediction
for AK mud, using the model of Jiang (1993). The measurements were made by observing the horizontal
displacement of an initially vertically injected dye streak adjacent to the flume side wall made of plexiglass. Two
sources of error are noteworthy in terms of the observed discrepancy between the data points and the model
calculated profile. Firstly, the data points incorporate the effects of drag close to the side wall and, secondly, the
calculated mass transport velocity profile is approximate due to analytical limitations (Jiang, 1993). Nevertheless,
the overall trends do compare favorably. Accurate field information on mass transport velocity is apparently sparse;
however, the model was applied to predict the rate of mass transport of transient mudbanks (of KI mud) off the
coast of Kerala (India). The predicted 0.22 km/d shoreward transport rate at the onset of monsoon at least appears
to be of the correct order of magnitude (Mathew, 1992; Jiang, 1993).
From an experiment meant to investigate mud dynamics, measured and predicted profiles of velocity and
dynamic pressure are compared in Fig. 31a,b for CK mud. The calculations are based on the model of Maa (1986),
in which the mud bottom was subdivided into four layers of thicknesses indicated, each characterized by density,
viscosity and shear modulus of elasticity. The dynamic pressure at first decreases, then increases with depth below
the interface. This variation has been explained by Maa (1986) in terms of the relative influences of the effective
viscous term and the pressure term in the equation of motion. Note that the model predicts an interfacial shear stress
0.20
0.40
0.60
0.80
0.10
MASS TRANSPORT VELOCITY (cm/s)
Figure 30. Mass transport velocity in AK mud. Wave amplitude 2 cm, frequency 1 Hz, h1 = 14 cm and mud
thickness h2 = 17 cm (after Jiang, 1993).
E
220
0
0
O10
0 10 20
VELOCITY AMPLITUDE,
(cm/s)
1.10
NORMALIZED PRESSURE
AMPLITUDE, (ga)
AMPLITUDE, (p/pyga)
Figure 31. a) Horizontal and vertical velocity amplitudes in water and CK mud; b) Dynamic pressure amplitude
profile corresponding to Fig. 3 1a. Smooth profiles have been drawn based on spatially discretized calculations
(after Maa and Mehta, 1987; Maa, 1986).
(amplitude, rb) of 0.19 Pa, which was too small to cause a substantial amount of interfacial erosion during the
experiment. Yet the bottom mud showed a measurable horizontal oscillation driven by the pressure gradient. This
type of a situation is likely to be ubiquitous in nature, with the result that under typically gentle, fair weather wave
action mud may oscillate, even while the upper water column is relatively free of turbidity, since the interfacial
stress is insufficient to cause significant mud scour. Thus, for example, in Lake Okeechobee, Florida, daily wave
induced mud oscillation tends to retain the top 520 cm layer of bottom mud in a fluidlike state, which in turn
presumably facilitates the transport of nutrients across the mudwater interface, and influences the trophic state of
this large and shallow lake (Jiang and Mehta, 1992).
5.2.3 Bed/Interface Erosion
The timevariation of the wavemean and depthmean dry density (concentration) of suspended sediment of a
bed of CK mud is shown in Fig. 32 in test T1 of Cervantes (1987) mentioned in Section 4.6. The normalized
parameters are: dry density, Ap = PD/DDS, and time, 0=(w/hl)t. Note the almost instantaneous rise of the
normalized dry density to 0.86, followed by a drop and subsequent attainment of a steady state (p,l). The
peculiar nature of the curve reflects the corresponding variation of with normalized time (t) shown in Fig. 20.
The function given in the inset of Fig. 20 has two components, the first of which reflects surface erosion and the
second is attributed to MohrCoulomb type bed failure or mass erosion. Initially the influence of the second term
is dominant in Fig. 32 and, as steady state approaches surface erosion (first term) becomes the main contributor.
Thus, qualitatively the response curve, i.e. the timevariation of pD, can be divided into the three phases shown
in Fig. 33. In phase PI mass erosion is dominant, PII is characterized by settling, and surface erosion is dominant
in PIII. The occurrence of the mass erosion peak may be of noteworthy significance in nature. Although supportive
field data are apparently sparse, such peaks are probably quite common at the time of comparatively sudden onset
of increased storm wave action, amounting to an episodic generation of turbidity.
The timevariation of the vertical profile of the dry density during an erosion experiment is shown in Fig. 34.
The bed was composed of HB (from Hillsboro Bay, Florida) mud, and simulation is via the model of Ross (1988),
which is based on the sediment transport equation given in Section 4.6. Note the rapid generation of the lutocline,
similar to what occurs in the field, e.g. Fig. 35 (adapted from Kemp and Wells, 1987). Of the four instantaneous,
vertical dry density profiles for suspended sediment shown in Fig. 35, one represents prefrontal wave conditions,
two were obtained during frontal passage, and the last is postfrontal. The data were obtained at a site on the eastern
margin of the Louisiana chernier plain where the tidal range is less than 0.5 m (Kemp and Wells, 1987). Wave
height during the winter cold front passage was on the order of 13 cm and period 7 s. Of particular interest is the
development of a fluid mud layer by the frontal windgenerated waves (profiles 2 and 3), which was previously
absent (profile 1). The postfrontal profile 4 further suggests that this layer may have persisted following front
passage, conceivably due to the comparatively low rate at which the layer was dewatered. The suspension density
in the upper water column was higher following the front than that during the front, possibly due to advection of
0
CQ
> 0.8
In
z
w
Q 0.6
LU
N
I
0.4
0.2
0.2 Test T1
(o = 6.3 rad/s)
Data
0 II
0 20 40 60
NORMALIZED DURATION, 0
Figure 32. Timevariation of normalized dry density (concentration), WD(t) = PD(t)/PDs, for CK mud, during test
T1. Mud depth (h2) was 16 cm, water depth (hi) was 17 cm, forcing wave amplitude (ao) was 3 cm and frequency
(f) was 1 Hz (after Cervantes, 1987).
PDs
Mass 1< Settling < Surface Erosion Dominant
Erosion Dominant
Dominant
0 ta tb TIME, t
Figure 33. Phases in the timevariation of PD(t), such as in Fig. 32 (after Cervantes, 1987).
.^ I I 1 I I11
S I Time Data Calculated
(hr)
m 0 25 5
0.25 .0 
2.0
I I 4.0 
C 0.20 i i
S.4 Mobile Suspension
0
< 0.15 Lutocline
< 0.10
0.10 I I I I i I I
102 101 100 101 102
DRY DENSITY (g/l)
Figure 34. Measured and modelcalculated dry density (concentration) profiles during erosion of Hillsboro Bay mud
(HB). Water depth (hi) 31 cm, mud depth (h) 12 cm, forcing wave amplitude, ao = 3 cm and wave frequency,
f = 1 Hz (after Ross, 1988).
100
S80
E
z
O
j= 60
j
LU
DRY DENSITY (g/I)
Figure 35. Vertical suspended sediment dry density (concentration) profiles obtained before, during, and after
the passage of a winter cold front at a coastal site in Louisiana. Times are relative to time of measurement of the
prefrontal profile 1 (adapted from Kemp and Wells, 1987).
suspended sediment from a neighboring area of high turbidity. Also note that in Fig. 34 as well as Fig. 35, the lack
of significant upward transport of sediment above the lutocline is consistent with the typically low waveinduced
mass diffusivities (Sheng, 1986).
5.3 LAKE OKEECHOBEE, FLORIDA
5.3.1 Setting
A large part of the central basin of Lake Okeechobee in southcentral Florida is overlain by a thin layer of dark,
organicrich (about 40% by weight) mud which is believed to be a significant storage for externally loaded
phosphorus. Figure 36 shows the mud thickness to be 80 cm at most, generally occupying the deeper part of the
lake. A typical variation of mud density with depth is shown in Fig. 37. As observed the top 10 cm of the mud
layer was comparatively low in density. Below this layer a rapid increase in density occurred and led to a relatively
uniformly dense mud underneath. The low density structure at the top is apparently characteristic of the organicrich
mud of comparatively uniform, fluidlike consistency (Hwang, 1989).
OKEECOBEE
& "77 \ o'z
r I~
Figure 36. Mud thickness contour map of Lake Okeechobee, Florida (after Kirby et al., 1989).
:::, pte.d '**0' v
S lAoohd VQaM doe. 9 LEOLADe
I nwa I
E 0
w
0
o
On 20
O Site
m 40 OK 22VC 0
4 OK 23 VC 0
I*
S1000 1100 1200
DENSITY p (kg/m3)
Figure 37. Typical mud bottom density profile based on vibrocore data from two nearby sampling sites (after
Hwang, 1989).
5.3.2 Mud Motion
In order to examine the nature of mud motion induced by wave action in the lake, an instrumented tower was
deployed at a site near Port Mayaca (Fig. 36). The mean water depth was 1.43 m and mud thickness 0.55 m. Water
surface variation was measured with a pressure transducer, and an electromagnetic current meter was used to
measure waveinduced water motion 87 cm below the mean water surface. To measure mud motion a small
accelerometer was embedded 20 cm below the mud surface where the density was 1,180 kg/m3. The shallow water
analytical model of Jiang (1993) described in Section 4.5.1 was used to aid in interpreting the data. In that model,
the water column is considered to be inviscid, and mud is assumed to be a highly viscous fluid. Mud viscosity was
measured in a rheometer; the kinematic viscosity at the density of 1,180 kg/m3 was found to be 1.76 x 102 m2/s,
which is very large compared to water (106 m2/s).
Figure 38a shows a typical wave spectrum obtained under moderate breeze (20 km/hr) from the westerly
direction. The dominant wave frequency was 0.42 Hz, with a significant height of 8 cm. It is noteworthy that
calculations (Mehta and Jiang, 1990) showed that at 20 km/hr wind speed, the wave height would have been four
fold larger over an otherwise comparable rigid bottom.
In Fig. 38b the measured horizontal water velocity amplitude spectrum is compared with that simulated by the
model using the wave spectrum of Fig. 38a as input. Given the model limitations, particularly with respect to the
hi 50
(a) Lake Okeechobee, Florida 
> 40
z
W 30
0
O 20
uz
u 10 
l oIii L
300 I i  i 
(b)
SE 240 
S240 / Simulation
0 "1
cI, I
W g180 '
03 i
S120 
> measurement
S 60
0.0 0.2 0.4 0.6 0.8 1.0
Z
0.20
S 0.10 
0 \\
0.0 0.2 0.4 0.6 0.8 1.0
FREQUENCY (Hz)
Figure 38. Data and model application, Lake Okeechobee, Florida: a) Measured water wave spectrum; b)
Comparison between measured and simulated water velocity spectra; c) Comparison between measured and
simulated mud acceleration spectra (after Jiang, 1993).
assumption of mud as a purely viscous fluid, and errors inherent in the data collection effort (Mehta and Jiang,
1990), the agreement between data and prediction appears to be acceptable.
Referring to Fig. 38c in which measured and simulated horizontal acceleration spectra in mud are compared,
the data as well as model simulation indicate the occurrence of measurable acceleration at about 0.04 Hz. This
behavior can be attributed to a forced second order wave that results from an interaction between different
frequencies of the forcing wind wave, as for example occurs in the case of surf beat over relatively flat open coast
beaches (Jiang and Mehta, 1992). Note that dominant seiching in Lake Okeechobee occurs at a much lower
frequency, on the order of 104 Hz. While energy contribution at 0.04 Hz frequency is also found in the wave
spectrum (Fig. 38a) and in the water velocity spectrum (Fig. 38b), the mud acceleration spectrum is enhanced at
the 0.04 Hz in comparison with the spectrum at the 0.42 Hz forcing frequency. This relative enhancement is due
to the strong frequency dependence of the wave attenuation coefficient, ki. Model results for example yielded ki =
0.0034 m1 at 0.42 Hz and 0.0013 m1 at 0.04 Hz. Note that if the bottom were rigid, ki would be 0(105)m1,
which is considerably smaller.
5.3.3 Resuspension
Using the vertical sediment mass transport numerical model of Ross (1988), the timeevolution of the vertical
dry density (concentration) profile was simulated under selected storm wave conditions. Profiles such as the one
in Fig. 37 were used to represent bed density variation with depth. For a given wave period and height, the bottom
stress amplitude was calculated from: rb=0.5 fPlu(hl,t)2, where f, is the wave friction factor (Jonsson, 1966).
The bottom velocity, u(h ,t), was obtained from the linear wave theory using wave amplitude, frequency and water
depth as input parameters. The neutral vertical mass diffusivity, ko, (Section 4.6) was obtained from
ko= c'a2w[sinh2k(z+hl)/2sinh2khl], based on the work of Hwang and Wang (1982), where a' is a free coefficient.
The k0 value was weighted by the damping parameter 4 (Section 4.6). Settling velocity data are given in Fig. 23.
In Fig. 39a the manner in which the suspension profile would evolve over a 48 hr period starting with a clear
water column is exemplified for a 6 s (w = 1.1 rad/s) storm wave of 1 m height in a 4.6 m deep water column
characteristic of the central part of the lake. Note the rather rapid formation of a fluid mud layer represented by
the lutocline. The layer thickness was found to be about 7 cm at 48 hr (Hwang, 1989). Note also however, that the
corresponding bed scour depth can be shown to be on the order of a centimeter only. Thus a very thin bed layer
generated an order of magnitude thicker fluid mud layer. Such layers, characterized by a significant density gradient,
have been observed in wave dominated environments (e.g. Fig. 35). Observe in Fig. 39a the very slow rise in the
suspension density in the upper water column due to characteristically low rates of upward mass diffusion. At 48
hr the dry density at the surface rose to only about 100 mg/l, very close to 102 mg/l reported by Gleason and Stone
(1975) in the southern part of the lake during a storm. If wave action were discontinued at 48 hr, the profiles of
Fig. 39b would result due to sediment settling under calm conditions. At 72 hr, or 24 hr after waves ceased, most
of the material is observed to have settled to the bottom, and the water column clarified.
5.4 SOUTHWEST COAST OF INDIA
An example of a drastic reduction in the wave energy due to wave passage over seasonal coastal mudbanks is
briefly noted here. Figure 40 shows the region offshore of the town of Alleppey in State of Kerala on the southwest
coast of India, where wave spectra were obtained at the two sites shown (Mathew, 1992). A schematic profile of
S 48 hr
X 2 I
D 24 hr
,UJ 12hr \\, i
6 hr %\%
S4
O ,Initial Bottom
I I I I I
103 102 101 100 101 102 103
SEDIMENT DRY DENSITY (g/l)
0 kI I I
S%~ 4 8 hr (b)
1 72 hr O
2
C(
0
J
w 3
W 4 
103 102 101 100 101 102 103
SEDIMENT DRY DENSITY (g/I)
Figure 39. a) Simulated timeevolution of suspension profile due to 0.9 m high, 4 s waves in a 4.6 m deep
water column; b) Settling of sediment once waves cease 11 hr from start of wave action (after Hwang, 1989).
the mudbank is shown in Fig. 41 (Nair, 1988), and the theological characteristics of the mud (KI) are given in Fig.
26 (and Table 9). The bottom beneath the mud elsewhere in this area is sandy. Thus the subaqueous bottom mud
is not always visible from the sandy beachface, even though the leading edge of the mudbank tends to occur quite
close to the shoreline. The mean depth of water at the offshore site was 10 m, and at the inshore site a little over
5 m. In fair weather the mudbank was well seaward of the offshore site, in deeper water. During the monsoon the
nearshore edge of the bank was shoreward of the inshore site, while the seaward edge of the bank was close to the
Figure 40. Coastal site off Alleppey in Kerala, India, where monsoonal, mudbanks occur. The pier is 300 m
long (after Mathew, 1992).
Peripheral Wave
Damping Zone High Water Line
Calm Water Area
Intense Wave Activity Mudbank Zone
Figure 41. Schematic profile of mudbank off the coast of Kerala, India (after Nair, 1988).
offshore site. Thus the bottom at the offshore site was devoid of mud. Figures 42a,b respectively show examples
of wave spectra in the absence and in the presence of 1 m thick mudbank at the inshore site. In fair weather the
energy reduction is observed to have been negligible. (Note the slight phase shift of the inshore spectrum relative
to offshore. This shift may be an artifact of the data analysis procedure.) In the monsoon case the energy reduction
between the two stations was about 85%. In fact, as a result of the mudbank the wave energy at the inshore site
was lower in monsoon than in fair weather, despite the occurrence of considerably more inclement offshore wave
activity during the monsoon. Not surprisingly, monsoonal mudbanks in Kerala have served as open coast havens
for small fishing vessels that would otherwise require sheltered harbors (Nair, 1988).
(a) (b)
a7/27/87 ( 7/1/89
d1
S0.8 Mudbank Absent Mudbank Present
Mathew (1992) E Mathew (1992)
1.0
0.6
/) U)
Offshore > 0.5
z z
S0.2 Inshore 
w Z
W 0.0 W 0.0
0.0 0.2 0.4 0.0 0.2 0.4
FREQUENCY,f (Hz) FREQUENCY,f (Hz)
Figure 42. Offshore and inshore wave spectra off Alleppey: a) without mudbank; b) with mudbank (after Mathew,
1992).
The model of Jiang (1993) was applied to the Alleppey data on a frequencybyfrequency basis by selecting
0.0025 m2/s for the eddy diffusivity, el, for the water column, a typical value (Lick, 1982), and taking the offshore
wave spectrum as input at x=0. Note that in Eqs. 50 and 51, vi=v is replaced by vl+ec to account for vertical
diffusion due to turbulence in the water column in a very approximate way. The fair weather case (Fig. 42a) was
treated by selecting arbitrarily high values of the rigidities G1 and G2 (applicable to Eq. 27 chosen as the constitutive
model for mud rheology), in order to simulate a rigid (sandy) bottom. For the (1 m thick) mud bottom case, values
of A2(f), Gi(f) and G2(f) for KI mud (0=0.12) were inputted (Mehta and Jiang, 1993). Results shown in Figs. 43a,b
indicate that the inshore wave spectra could be simulated reasonably well for the fair weather (Fig. 43a) as well as
monsoon (Fig. 43b) scenarios. As noted previously the model yielded a depthmean residual mud velocity (Stokes'
drift) of 0.22 km/day, which seems to be consistent with the timescale of mud motion at the onset of monsoon,
even though no detailed mud velocity measurement are available for comparison purposes (Mathew, 1992).
5.5 MOBILE BERM, ALABAMA
The sediment placement site for the Mobile underwater berm off Dauphin Island in Alabama, created as a
national demonstration project to highlight the beneficial role (wave height reduction in this case) of dredged
material, is shown in Fig. 44. The material was derived from the ship channel within Mobile Bay (McLellan et al.,
1990; Hands, 1990). Examples of offshore/inshore wave spectra are given in Figs. 45a,b under two different
offshore wave conditions. Wave energy reductions were significant, 29% and 46% respectively. Using a fixedbed
hydrodynamic model McLellan et al. (1990) showed that assuming a rigid (i.e. fixed bed) berm crest produced
negligible wave energy dissipation. It is believed that as a result of the shallow depth of water, on the order of
E 0.4 (a) Alleppey, India
^> ^Measurement
u 0.3
z
S\\ Simulation
> 0.2
LU
Z 0.1
LU
> 0.0\
S 0.0 0.1 0.2 0.3 0.4
FREQUENCY (Hz)
or (b)
E 0.4 (b) Alleppey, India
C 0.3
z
IJ
> 0.2
z 0.1
> 0.0 I I
0.0 0.1 0.2 0.3 0.4
FREQUENCY (Hz)
Figure 43. Comparison between measured and modelsimulated inshore wave spectra off Alleppey in Kerala, India:
a) fair weather condition (mudbank absent); b) monsoonal condition (mudbank present).
6.5 m over the berm, the main cause of damping is energy absorption by the deposited mud. Diver observations
at the site suggested the occurrence of surface waveforced interfacial mud waves propagating along the compliant
crest (McLellan, personal communication). Such a movement can be construed as a manifestation of the participation
of the bottom material in the energy dissipation process. Furthermore, the high degree of stability of the berm
against displacement and deterioration pointed to the fact that wave action became sufficiently weak over the berm
to prevent significant scour of the berm (Dredging Research Technical Notes, 1992).
The wavemud interaction model of Jiang (1993) was used to simulate the inshore wave spectra at the Mobile
berm site, given measured offshore spectra in Figs. 45a,b. The computations were done on a frequencyby
frequency basis. A 6 m berm height (=h2) and the 6.5 m water depth (=hi) were selected as representative vertical
dimensions. In general, the effective berm width for calculating the degree of wave attenuation would depend on
the direction of wave approach; for the present purpose a width of 140 m was selected, giving some allowance for
Figure 44. Construction site (2,750 m long corridor for dredged material placement) for the Mobile berm, and
offshore/inshore sites of wave measurement (after McLellan et al., 1990).
the berm slopes. Thus dissipation was assumed to occur over this distance only. Rheological parameters for the mud
(MB) are given in Table 9 (for 0= 0.11, corresponding to the in situ mud density).
Comparisons between simulated and measured inshore spectra are given in Figs. 46a,b. The simulated spectral
energy is observed to be generally in agreement with the measurement at low frequencies, but is consistently lower
than measured for frequencies exceeding about 0.25 Hz, i.e. periods smaller than 4 s. In that context it must be
noted that, as a rule, the output spectrum from the model will have the same general shape as the offshore one,
which also is the case here. Compare, for example, the measured offshore spectrum with the simulated inshore one
in Fig. 46a. These two are generally similar in shape. Next compare the measured inshore spectrum with the
measured offshore one in the same figure. Note that for frequencies greater than about 0.30 Hz the measured
inshore wave energy was actually slightly higher than that at the offshore location. This feature suggests that the
source of the high frequency inshore waves may have been at least partly different from that represented by the
measured offshore spectrum. In any event, diagnostic applications such as this one provide a basis for developing
design guidelines for wave absorbing berms in future. A hypothetical design example has been discussed by Mehta
and Jiang (1993).
Sand Island
II Mobile
SPoint
II
II
Inshore If
/ II
oIl
Offshore W /it
3 II
II
0.25
0.20 
0.15
0.10 
0.05 
McLeIIan et a
McLellan et a
Hmax =0.9 m
Oftshore
\I
I\
Inshore
FREQUENCY (Hz)
FREQUENCY (Hz)
Figure 45. Offshore and inshore wave spectra at the Mobile berm site for two different wave conditions at the
offshore site characterized by the maximum wave height, Hmx: (a) Hma = 0.9 m, and (b) Hmx = 1.5 m (after
McLellan et al., 1990).
I. (1990)
0.1 0.2 0.3
FREQUENCY (Hz)
0.4 0.5
Figure 46. Comparison between measured and modelsimulated inshore wave spectra at the Mobile berm site
corresponding to two offshore wave conditions: (a) Hnax = 0.9 m (see Fig. 45a), and (b) Hmax = 1.5 m (see Fig.
45b). Measured offshore spectra have been included for reference.
0.8
0.6
0.4
0.2
0.0
0.0
BIBLIOGRAPHY
Adams C.E., Wells J.T. and Park Y.A., 1993. Wave motions on a lutocline above a stably stratified bottom
boundary layer. In.: Nearshore and Estuarine Cohesive Sediment Transport, A.J. Mehta ed., American
Geophysical Union, Washington, DC, 393410.
Alishahi M.R. and R.B. Krone, 1964. Suspension of cohesive sediments by windgenerated waves. Technical Report
HEL29, Hydraulic Engineering Laboratory, University of California, Berkeley, 24p.
Allen R.H., 1972. A glossary of coastal engineering terms. Miscellaneous Paper 272, U.S. Army Corps of
Engineers Coastal Engineering Research Center, Fort Belvoir, VA, 55p.
Barnes H.A., Hutton J.F. and Walters K., 1989. An Introduction to Rheology, Elsevier, Amsterdam, 208p.
Cervantes E.E., 1987. A laboratory study of fine sediment resuspension by waves. M.S. Thesis, University of
Florida, Gainesville, 64p.
Chou H.T., 1989. Rheological response of cohesive sediments to water waves. Ph.D. Dissertation, University of
California, Berkeley, 149p.
Conners D.H., Risenberg F., Charney R.D., McEwen M.A., Krone R.B. and Tchobanoglous G., 1990. Research
needs: salt marsh restoration, rehabilitation, and creation techniques for Caltrans construction projects. Report
prepared for California Department of Transportation by the Civil Engineering Department, University of
California, Davis, 61p.
Cueva I.P., 1993. On the response of a muddy bottom to surface water waves. Journal of Hydraulic Research,
31(5), 681696.
Dalrymple R.A. and Liu P.L.F., 1978. Waves over soft muds: a twolayer fluid model. Journal of Physical
Oceanography, 8, 11211131.
Dawson T.H., 1978. Wave propagation over a deformable sea floor. Ocean Engineering, 5, 227234.
Day P.R. and Ripple C.D., 1966. Effect of shear on suction in saturated clays. Soil Society of America Proceedings,
30, 625679.
Dean R.G. and Dalrymple R.A. 1984. Water Wave Mechanics for Engineers and Scientists, PrenticeHall,
Englewood Cliffs, NJ, 353p.
Dredging Research Technical Notes, 1992. Monitoring Alabama berms. DRP108, U.S. Army Engineer Waterways
Experiment Station, Vicksburg, MS, 14p.
Feng J., 1992. Laboratory experiments on cohesive soil bed fluidization by water waves. M.S. Thesis, University
of Florida, Gainesville, 109p.
Foda M.A., 1989. Sideband damping of water waves over a soft bed. Journal of Fluid Mechanics, 201, 189201.
Foda M.A., Tzang S.Y. and Maeno Y., 1991. Resonant soil liquefaction by water waves. Proceedings of Geo
Coast'91, 1, Port and Harbor Research Institute, Yokohama, Japan, 549583.
Foda M.A., Hunt J.R. and Chou H.T., 1993. A nonlinear model for the fluidization of marine mud by waves.
Journal of Geophysical Research, 98(C4), 70397047.
Gade H.G., 1957. Effects of a nonrigid, impermeable bottom on plane surface waves in shallow water. M.S.
Thesis, Texas A & M University, College Station, 35p.
Gade H.G., 1958. Effects of a nonrigid, impermeable bottom on plane surface waves in shallow water. Journal
of Marine Research, 16(2), 6182.
Gleason P.J. and Stone P.A., 1975. Prehistoric trophic level status and possible cultural influences on the
enrichment of Lake Okeechobee. Unpublished Report, South Florida Water Management District, West Palm
Beach, 133p.
Hands E.G., 1990. Results of monitoring the disposal berm at Sand Island, Alabama, report 1: construction and
first year's response. Technical Report DRP902, U.S. Army Engineer Waterways Experiment Station,
Vicksburg, MS, 59p.
Hsiao S.V. and Shemdin O.H., 1980. Interaction of ocean waves with a soft bottom. Journal of Physical
Oceanography, 10, 605610.
Horikawa K., 1978. Coastal Engineering, Wiley, New York, 412p.
Hunt L.M. and Groves D.G., 1965. A Glossary of Ocean Science and Undersea Technology Terms. Compass
Publications, Arlington, VA, 180p.
Hwang K.N., 1989. Erodibility of fine sediment in wavedominated environments. M.S. Thesis, University of
Florida, Gainesville, 159p.
Hwang P.A. and Wang H., 1982. Wave kinematics and sediment suspension at wave breaking point. Technical
Report No. 13, Department of Civil Engineering, University of Delaware, Newark, 173p.
Inglis C.C. and Allen F.H., 1957. The regimen of the Thames estuary as affected by currents, salinities, and river
flow. Proceedings of the Institution of Civil Engineers, London, 7, 827868.
Isobe M., Huynh T.N. and Watanabe A., 1992. A study on mud mass transport under waves based on an empirical
rheology model. Proceedings of the 23rd Coastal Engineering Conference, 3, ASCE, New York, 30933106.
Jackson J.R., 1973. A model study of the effects of small amplitude waves on the resuspension of finegrain
cohesive sediments. M.S. Thesis, University of New Hampshire, Durham, 53p.
James A.E., Williams D.J.A. and Williams P.R., 1988. Small strain, low shear rate rheometry of cohesive
sediments. In: Physical Processes in Estuaries, J. Dronkers and W. van Leussen eds., SpringerVerlag, Berlin,
488500.
Jiang F., 1993. Bottom mud transport due to water waves. Ph.D. Dissertation, University of Florida, Gainesville,
222p.
Jiang F. and Mehta A.J., 1992. Some observations on fluid mud response to water waves. In: Dynamics and
Exchanges in Estuaries and the Coastal Zone, D. Prandle ed., American Geophysical Union, Washington, DC,
351376.
Jiang L. and Zhao Z., 1989. Viscous damping of solitary waves over fluidmud seabeds. Journal of Waterway,
Port, Coastal, and Ocean Engineering, 115(3), 345362.
Jiang L., Kioka W. and Ishida A., 1990. Viscous damping of cnoidal waves over fluidmud seabed. Journal of
Waterway, Port, Coastal and Ocean Engineering, 116(4), 470491.
Jonsson I.G., 1966. Wave boundary layer and friction factors. Proceedings of the 10th Coastal Engineering
Conference, ASCE, New York, 127148.
Keedwell M.J., 1984. Rheology and Soil Mechanics. Elsevier, London, 339p.
Kendrick M.P. and Derbyshire B.V., 1985. Monitoring of a nearbed turbid layer. Report SR44, Hydraulics
Research, Wallingford, U.K., 20p.
Kemp G.P. and Wells J.T., 1987. Observations of shallowwater waves over a fluid mud bottom: implications to
sediment transport. Proceedings of Coastal Sediments'87, 1, ASCE, New York, 363378.
Kirby R.R., Hobbs C.H. and Mehta A.J., 1989. Fine sediment regime of Lake Okeechobee, Florida. Report
UFL/COEL89/009, Coastal and Oceanographic Engineering Department, University of Florida, Gainesville,
77p.
Krizek R.J., 1971. Rheological behavior of clay soils subjected to dynamic loads. Transactions of the Society of
Rheology, 15(3), 433489.
Krone R.B., 1962. Flume studies of flocculation as a factor in estuarial shoaling processes. Final Report, Hydraulic
Engineering Laboratory and Sanitary Engineering Research Laboratory, University of California, Berkeley,
118p.
Krone R.B., 1979. Sedimentation in San Francisco Bay system. In: The Urbanized Estuary, California Academy
of Sciences, San Francisco, 8596.
Lamb T.W. and Whitman R.V., 1969. Soil Mechanics. Wiley, New York, 563p.
Lick W., 1982. Entrainment deposition and transport of finegrained sediment in lakes. Hydrobiologia, 91, 3140.
Liu K.F. and Mei C.C., 1989. Effects of waveinduced friction on a muddy seabed modelled as a Binghamplastic
fluid. Journal of Coastal Research, 5(4), 777789.
Maa P.Y., 1986. Erosion of soft mud by waves. Ph.D. Dissertation, University of Florida, Gainesville, 296p.
Maa, P.Y. and Mehta A.J., 1987. Mud erosion by waves: a laboratory study. Continental Shelf Research,
7(11/12), 12691284.
Maa P.Y. and Mehta A.J., 1989. Considerations on soft mud response under waves. In: Estuarine Circulation,
B.J. Neilson, A. Kuo, and J. Brubaker, eds., Humana Press, Clifton, NJ, 309336.
Maa P.Y. and Mehta A.J., 1990. Soft mud response to water waves. Journal of Waterway, Port, Coastal, and
Ocean Engineering, 116(5), 634650.
MacPherson H., 1980. The attenuation of water waves over a nonrigid bed. Journal of Fluid Mechanics, 97(4),
721742.
Mallard W.W. and Dalrymple R.A., 1977. Water waves propagating over a deformable bottom. Proceedings of
the Offshore Technology Conference, OTC 2895, Houston, 141146.
Mathew J., 1992. Wavemud interaction in mudbanks. Ph. D. Dissertation, Cochin University of Science and
Technology, Cochin, Kerala, India, 139p.
McCave I.N., 1971. Waveeffectiveness at the sea bed and its relationship to bedforms and deposition of mud.
Journal of Sedimentary Petrology, 41(1), 8996.
McLellan T.N., Pope M.K. and Burke C.E., 1990. Benefits of nearshore placement. Proceedings of the 3rd Annual
National Beach Preservation Technology Conference, Florida Shore and Beach Preservation Association,
Tallahassee, 339353.
Mehta A.J., 1988. Laboratory studies on cohesive sediment deposition and erosion. In: Physical Processes in
Estuaries, J. Dronkers and W. van Leussen eds., SpringerVerlag, Berlin, 427445.
Mehta A.J., 1989. On estuarine cohesive sediment transport. Journal of Geophysical Research, 94(C10), 14303
14314.
Mehta A.J., 1990. Understanding fluid mud in a dynamic environment. GeoMarine Letters, 11, 113118.
Mehta A.J., 1991a. Strategy for fine sediment regime investigation: Lake Okeechobee, Florida. Proceedings of the
International Conference on Coastal and Port Engineering in Developing Countries, Mombasa, Kenya, 303
917.
Mehta A.J., 199 b. Characterization of cohesive soil bed surface erosion, with special reference to the relationship
between erosion shear strength and bed density. Report UFL/COEL/MP91/4, Coastal and Oceanographic
Engineering Department, University of Florida, Gainesville, 83p.
Mehta A.J., 1991c. Review notes on cohesive sediment erosion. Proceedings of Coastal Sediment'91, 1, ASCE,
New York, 4053.
Mehta A.J. and Maa P.Y., 1986. Waves over mud: modeling erosion. Proceedings of the Third International
Symposium on River Sedimentation, University of Mississippi, University, 588601.
Mehta A.J. and Srinivas R., 1993. Observations on the entrainment of fluid mud by shear flow. In: Nearshore and
Estuarine Cohesive Sediment Transport, A.J. Mehta ed., American Geophysical Union, Washington, DC, 224
246.
Mehta A.J. and Jiang F., 1990. Some field observations on bottom mud motion due to waves. Report UFL/COEL
90/008, Coastal and Oceanographic Engineering Department, University of Florida, Gainesville, 85p.
Mehta A.J. and Jiang F., 1993. Some observations on water wave attenuation over nearshore underwater mudbanks
and mud berms. Report UFL/COEL/MP93/01, Coastal and Oceanographic Engineering Department,
University of Florida, Gainesville, 45p.
Mehta A.J. and Lee S.C., 1994. Problems in linking the threshold condition for the transport of cohesionless and
cohesive sediment grains. Journal of Coastal Research (in press).
Mei C.C. and Liu K.F., 1987. A Bingham plastic model for muddy sea bed under long waves. Journal of
Geophysical Research, 92(C13), 1458114594.
Migniot C. and Hamm, L., 1990. Consolidation and theological properties of mud deposits. Proceedings of the
22nd Coastal Engineering Conference, 3, ASCE, New York, 29752983.
Mimura N., 1993. Rates of erosion and deposition of cohesive sediments under wave action. In: Nearshore and
Estuarine Cohesive Sediment Transport, A.J. Mehta, ed., American Geophysical Union, Washington, DC, 247
264.
Mindlin R.D. and Deresfewicz H., 1953. Elastic spheres in contact under varying oblique forces. Journal of Applied
Mechanics, 20, 327344.
Nair A.S.K., 1988. Mudbanks (chakara) of Kerala a marine environment to be protected. Proceedings of the
National Seminar on Environmental Issues, University of Kerala Golden Jubilee Seminar, Kerala, India, 7693.
National Research Council, 1989. Using Oil Spill Dispersants on the Sea. Committee on the Effectiveness of
Oil Spill Dispersants, Marine Board, National Academy Press, Washington, DC, 352p.
National Research Council, 1994. Protecting and Restoring Marine Habitat: The Role of Engineering and
Technology. Committee on the Role of Technology in Marine Habitat Protection and Enhancement, Marine
Board, National Academy Press, Washington, DC (under preparation).
Nichols M.M., 1985. Fluid mud accumulation processes in an estuary. GeoMarine Letters, 4, 171176.
Odd N.M.V. and Rodger J.G., 1986. An analysis of the behavior of fluid mud in estuaries. Report SR84,
Hydraulics Research, Wallingford, U.K., 25p.
Parchure T.M. and Mehta A.J., 1985. Erosion of soft cohesive sediment deposits. Journal of Hydraulic
Engineering, 111(10), 13081326.
Parker W.R. and Kirby R., 1982. Time dependent properties of cohesive sediment relevant to sedimentation
management a European experience. In: Estuarine Comparisons, V.S. Kennedy ed., Academic Press, San
Diego, 573590.
Partheniades, E., 1977. Unified view of wash load and bed material load. Journal of the Hydraulics Division of
ASCE, 103(9), 10371057.
Ross M.A., 1988. Vertical structure of estuarine fine sediment suspensions. Ph.D. Dissertation, University of
Florida, Gainesville, 206p.
Ross M.A., Lin C.P. and Mehta A.J., 1987. On the definition of fluid mud. Proceedings of the 1987 National
Conference on Hydraulic Engineering, ASCE, New York, 231236.
Ross M.A. and Mehta A.J., 1989. On the mechanics of lutocline and fluid mud. Journal of Coastal Research, SIS,
5161.
Ross M.A. and Mehta A.J., 1990. Fluidization of soft estuarine mud by waves. In: The Microstructure of Fine
Grained Sediments: From Mud to Shale, R.H. Bennett ed., SpringerVerlag, New York, 185191.
Sakakiyama T. and Bijker E.W., 1989. Mass transport velocity in mud layer due to progressive waves. Journal of
Waterway, Port, Coastal, and Ocean Engineering, 115(5), 614633.
Scarlatos P.D. and Mehta A.J., 1993. Instability and entrainment mechanisms at the stratified fluid mudwater
interface. In: Nearshore and Estuarine Cohesive Sediment Transport, A.J. Mehta, ed., American Geophysical
Union, Washington, DC, 205223.
Schreuder F.W.A.M., Van Dieman A.J.G. and Stein H.N., 1986. Viscoelastic properties of concentrated
suspensions. Journal of Colloid and Interface Science, 111(1), 3543.
Schuckman B. and Yamamoto T., 1982. Nonlinear mechanics of seabed interactions part II wave tank
experiments on water wave damping by motion of clay beds. Technical Report TR821, Rosenstiel School of
Marine and Atmospheric Science, University of Miami, 132p.
Sheng Y.P., 1986. Finitedifference models for hydrodynamics of lakes and shallow seas. In: Physicsbased
modeling of lakes, reservoirs and impoundments, G. William ed., ASCE, New York, 146228.
Sheng Y.P., Cook V., Peene S., Eliason D., Wang P.F. and Schofield S., 1986. A field and modeling study of
fine sediment transport in shallow waters. In: Estuarine and Coastal Modeling, M.L. Spalding ed., ASCE,
New York, 113122.
Shibayama T., Takikawa H. and Horikawa K., 1986. Mud mass transport due to waves. Coastal Engineering in
Japan, 29, 151161.
Shibayama T., Aoki T. and Sato S., 1989. Mud mass transport rate due to waves: a viscoelastic model. Proceedings
of the 23rd IAHR Congress, Ottawa, Canada, B567B574.
Shibayama T., Okuno M. and Sato S., 1990. Mud transport rate in mud layer due to wave action. Proceedings of
the 22nd Coastal Engineering Conference, 3, ASCE, New York, 30373049.
Smith J.D., 1977. Modeling of sediment transport on continental shelves. In: The Sea, Vol. 6, E.D. Goldberg et
al. eds., Wiley, New York, 539577.
Smith T.J. and Kirby R., 1984. Generation, stabilization and dissipation of layered fine sediment suspensions.
Journal of Coastal Research, SI5, 6373.
Somly6dy L., Herodek S. and Fischer J., editors, 1983. Eutrophication of Shallow Lakes: Modeling and
Management, the Lake Balaton Case Study. International Institute for Applied Systems Analysis, Laxenburg,
Austria, 377p.
Sterling G.H. and Strohbeck E.E., 1973. The failure of the South Pass 70 "B" platform in hurricane Camile.
Proceedings of the 5th Offshore Technology Conference, Houston TX, 719730.
Suhayda J.N., 1986. Interaction between surface waves and muddy bottom sediments. In: Estuarine Cohesive
Sediment Dynamics, A.J. Mehta, ed., SpringerVerlag, Berlin, 401428.
Thiers G.R. and Seed H.B., 1968. Cyclic stressstrain characteristics of clays. Journal of the Soil Mechanics and
Foundations Division of ASCE, 92(2), 555569.
Thimakorn P., 1980. An experiment on clay suspension under water waves. Proceedings of the 17th Coastal
Engineering Conference, 3, ASCE, New York, 28942906.
Thimakorn P., 1984. Resuspension of clays under waves. In: Seabed Mechanics, B. Denness ed., Graham and
Trotman, London, 191196.
Tsuruya H., Nakano S. and Takahama J., 1987. Interactions between surface waves and a multilayered mud bed.
Report of the Port and Harbor Research Institute, Ministry of Transport, Japan, 26(5), 137173.
Tsuruya H., Murakami K. and Irie I., 1990. Numerical simulation of mud transport by a multilayered nested grid
model. Proceedings of the 22nd Coastal Engineering Conference, 3, ASCE, New York, 29983011.
Tubman M. and Suhayda J.N., 1976. Wave action and bottom movements in fine sediments. Proceedings of the
15th International Coastal Engineering Conference, ASCE, New York, 11681183.
Tver D.F., 1979. Ocean and Marine Dictionary. Cornell Maritime Press, Centerville, MD, 367p.
Wang H. and Xue H.C., 1990. ChinaU.S. joint muddy coast research, part 1, a review of hydrological and
sedimentary processes in Hangzhou Bay, China. Report UFL/COEL90/014, Coastal and Oceanographic
Engineering Department, University of Florida, Gainesville, 60p.
Wells J.T., 1983. Dynamics of coastal fluid muds in low, moderate, and hightiderange environments. Canadian
Journal of Fisheries and Aquatic Sciences, 40(1), 130142.
Wells J.T. and Kemp G.P., 1986. Interaction of surface waves and cohesive sediments: field observations and
geologic significance. In: Estuarine Cohesive Sediment Transport, A.J. Mehta ed., SpringerVerlag, Berlin,
4365.
Williams D.J.A. and Williams P.R., 1992. Laboratory experiments on cohesive soil bed fluidization by water
waves, part II, in situ rheometry for determining the dynamic response of bed. Report UFL/COEL92/O15,
Coastal and Oceanographic Engineering Department, University of Florida, Gainesville, 24p.
Wolanski E., Chappell J., Ridd P. and Verbessy R., 1988. Fluidization of mud in estuaries. Journal of Geophysical
Research, 93(C3), 23512361.
Wolanski E., Gibbs R., Ridd P. and Mehta A., 1992. Settling of oceandumped dredged material, Townsville,
Australia. Estuarine, Coastal and Shelf Science, 35, 473489.
Yamamoto T., 1982. Nonlinear mechanics of water waves interactions with sediment beds. Applied Ocean
Research, 4(2), 99106.
Yamamoto T., 1983. On the response of a Coulombdamped poroelastic bed to water waves. Marine Technology,
5(2), 93130.
Yamamoto T., Koning H.L., Sellmeijer H. and Van Hijum E., 1978. On the response of the poroelastic bed to
water wave. Journal of Fluid Mechanics, 87(1), 193206.
Yamamoto T. and Takahashi S., 1985. Wave damping by soil motion. Journal of Waterway, Port, Coastal, and
Ocean Engineering, 111(1), 6277.
APPENDIX A: EXPLANATORY NOTES FOR TABLE 7
Note 1: The solution is based on a twofluid system consisting of an inviscid fluid overlying a lower viscous mud
layer in laminar flow, which is assumed to be infinite in depth, driven by linear water waves. A boundary layer
approach via prescription of a boundary layer correction to the velocity potential in the mud region is then adopted
whereby mud flow is assumed inviscid except in the boundary layer region. Two wave modes emerge from the
solution. The first mode, w2=gkr, which is only true for infinitely deep mud layer, precludes a discontinuity of
horizontal velocity across the interface and, hence, there is no boundary layer formation and no associated damping.
For the second mode, w2/gkr=[(P2/Pl)l]tanhkrhl/[(P2/Pl)+tanhkrh1], the wavemean rate of viscous energy
dissipation per unit area in the boundary layer is computed from:
= P2 U2 1 2d I [ 2v2 11/2 2kh, (w2 2 2] (A.1)
ED = P922 a dz = W / e 2 (gkr) a
The relative dominance of the two wave modes is decided by the nature of wave generation (Dean and Dalrymple,
1984). While the first mode is more likely under the circumstance of waves propagating into a muddy region, the
second mode will dominate if the waves are generated at the interface by a displacement of the mud.
Note 2: The solution is based on a twolayered viscous fluid system driven by linear water waves propagating in
water of constant depth, hi, overlying a viscous mud layer of thickness, h2, with appropriate boundary layer
approximation to render the solution analytical. These solutions apply when the parameters, e = (i2/g)(iw) 112, i= 1,
2, are small whence the motion is essentially irrotational except near boundaries where the viscous boundary layer
thickness is on the order of (21/w)1/2, i= 1,2. Hence, the solutions are valid for large values of [w/(22)]1/2h2, i.e.,
the mud depth is large relative to the wave boundary layer thickness. Assuming that the free surface is
uncontaminated, energy dissipation occurs only in the boundary layers, and is partitioned as follows:
CD1 ii" 1 1 dz (A.2)
hti 7 I
CD2 = P2"2. L J2 dz (A.3)
D3 = P2'2 f dz (A.4)
(h +h2)
where ui, i= 1,2,3 denote the rotational velocity components in the direction of wave propagation in the respective
boundary layers, CDI is the mean energy dissipation rate in the water boundary layer near the interface, CD2 is the
mean energy dissipation rate in the mud boundary layer near the interface and CD3 is the mean energy dissipation
rate in the mud boundary layer near the rigid bottom, overbar denotes time averaging and 00" implies "outside"
the boundary layer. The various terms are given by:
u1 = C1exp{(1 +i)[w/(2v1)]112(z+ hl)}exp[i(kxwt)]
u= C2exp{(1 +i)[w/(2v2)]1/2(z+ h)}exp[i(kxwt)]
3= C3exp{(1 +i)[w/(2v2)11/2(z+hl +h2)}exp[i(kxwt)],
Cl= {[gka/(wsinhkhl)][(p2/pl)( 2/ l)1/2]l[1 + (p2/Pl)( 2 1)l/2]}{ (2/gk)
[(1/2)sinh2khl](cothkh1 +cothkh2)[(w2/gk)tanhkhl)]},
C2=(Pl/P2)(vl/v2)1/2Cl
C3= {gka/wcoshkhl[(w2/gk)tanhkhl]}/sinhkh2
and the wave number is solved iteratively from:
w2/gk = [l(124mn)12]/(2m)
where
l=(p2/Pl)tanhk(h1 +h2)(1 +tanhkhltanhkh2)
m= (p2/P) +tanhkhtanhkh2
n= [(p2/p)l]tanhkh tanhkh2
Note 3: The rate of wave energy dissipation is computed by assuming mud as a viscous medium (Dean and
Dalrymple, 1984):
CD = i 12 u]2 + dz (A5)
0 ax ax aZ
where the overbar denotes wavemean value, and hi and h2 denote the depths of water and mud layers, respectively.
A shallow water wavemud interaction model (Jiang and Mehta, 1993), which describes a twolayered mud/water
system forced by progressive surface waves of periodicity specified by wave frequency, w, is used to compute the
required velocity gradient inputs.
Since the model assumes an inviscid upper water layer on the premise that pressure and inertial forces are
typically dominant in governing water motion and an irrotational flow field is an apt description here, wave
dissipation in the water layer is nil. Furthermore, by ignoring the velocity gradient in the vertical (shallow water
assumption), the above equation simplifies to:
 P2 2 2 2 + 2u 2d (A.6)
which may be conveniently divided into two terms,
2 x U] hj2 'u (A.7)
'D DI+ D2 P2, 2 2 dz +P2V2 [ dz
The appropriate final expressions are (Feng, 1992):
D' = P22 i 2 2 A
(A.8)
Ao 2 (1+ A0)1
(B 1) Cc + C1
2m m 4m II
h1 (A (2A 6 I f1) (1 + A 1 (A.9)
2 pm2p [Fr ) 2 2 + 4m 2m
where A=a/h1, AO = tanh(mh2), B0 = cosh(mh2), CO = sinh(mhz), B1 = cosh(2mh2), C1 = sinh(2mh2),
h2 = h2/hi, m = (iRe)1/, i=1)12, and Re = whl2/v2 is the wave Reynolds number, k = khI is the
normalized complex wave number given by:
SI +fi2r [(1 +i2r) 4rf2r]/2 1/2 (A.10)
Fr 2rfi2r
where k=kr+iki, kr = wave number, ki = wave attenuation coefficient, r=(p2P1)/p2, r= 1tanh(mh2)/(mh2) and
Fr=w(h1/g)1/2 is the wave Froude number.
Note 4: The solution is based on a system consisting of linear water waves propagating over a nonrigid bed
characterized as a Voigt viscoelastic body overlain by inviscid water, with appropriate assumptions to obtain an
explicit expression for wave attenuation. By introducing a viscoelastic parameter, v*= y =v2+iG/wp2, where G
is the shear modulus of elasticity, it is shown that the equations of motion for a viscoelastic medium can be reduced
to the form of the linearized NavierStokes equations for a viscous fluid. The following assumptions are then
introduced to render the solutions explicit:
a) Oscillations of the bed due to the propagating waves are small in comparison with the surface waves, as
in the case of an almost rigid bed, or mathematically, I k2v/w I > 1 where k is the complex wave number
defined in Note 3;
b) Mud region is assumed to be infinite in depth;
c) Wave length is assumed long compared to water depth.
The resulting expression for k, after ignoring higher order terms, becomes:
SPlSgG PigV2
k +gG + ip+gV2 (A.11)
2 2 22 2 2 2
ghi 4P2 2(v2+G /p2w2) 4P2Z(P2+G2/p2W )
The mean rate of energy dissipation is then computed based on the mean work done on the lower layer per
unit area of the interface as:
E D plw dt (A.12)
TD=i
on z='r2=0 after linearizing, where T = wave period, and p, and wI are the pressure and vertical velocity,
respectively, in water. Upon timeaveraging and integration, the following expression results:
I = 1 2 w2(gkcoshkhi sinhkh) (A.13)
I m gk(gksinhkh, 2coshkhw)
where Ar = 2l pgG+ip, p2gv2 is defined as the amplitude ratio.
2 4p o2(v2 +G2/p22)
Note 5: The dissipation mechanism employed here is that due to Yamamoto and Takahashi (1985), attributed to
waveinduced motion in the soil bed. Based on a theoretical analysis for two elastic spheres, Mindlin and
Deresfewicz (1953) found that the nonlinear elasticity and the energy dissipation of granular materials, e.g. soils,
are due to Coulomb friction and are independent of the forcing frequency. In the frequency domain, the constitutive
relation is approximately given by the linear model, r = Gy, where 7 = shear stress, y = shear strain and
G = G(1 +ib) is the complex shear modulus. The real part, G, is the dynamic shear modulus, and 6 is the specific
loss given by = AW/2rW, in which AW = energy loss per cycle and W is the maximum elastic energy given by
W = 0.5G7'2, where y' = shear strain amplitude, both being expressed per unit area. The effect of soil Coulomb
friction on the attenuation of water waves is approximately estimated by the analytical solution obtained by
Yamamoto et al. (1978) for the quasistatic response of a homogeneous halfspace of porous elastic medium to water
waves. It is assumed that the soil moduli, G and 6, are constant throughout the bed. It is noted that the pure shear
strain condition is induced in the bed by the water wave for the entire phase. The magnitudes of the shear stress,
rm, and the shear strain, ym, are only a function of depth, z, and are given by Tm = p0okzekz and
ym = (P0/G)kozekz, respectively. Here pg is the amplitude of bottom pressure given by po=plga(l/coshkohl),
which is the waveinduced dynamic pressure component based on linear wave theory.
The energy loss, AW, due to soil Coulomb friction is given by:
AW= 27r J rmmdz (A.14)
which after integration becomes:
AW Ir (Pig)2a2S (A.15)
2 koGcosh2k0hl
where ko is the wave number for a rigid bottom. Accordingly, the rate of energy dissipation becomes:
AW = (18g)2 wSa2 (A.16)
CD= =
T 4 Gkocosh2k0h
Note 6: Here the approximate analysis assuming quasistatic motion of homogeneous soil bed (see Note 5) is
improved upon by utilizing the dynamic theory for the case of a homogeneous half space of Coulombdamped
poroelastic medium to account for the coupling of water waves and the bed motion (Yamamoto, 1982; 1983). While
two separate expressions for wave attenuation coefficient, due to soil Coulomb friction and percolation, respectively,
are obtained, only the former is used here based on the premise that typical permeability in mud is very small.
Substituting into the steady state wave energy equation for a constant water depth, CgdE/dx=CD, and assuming an
exponential decay law then leads to:
P2 aa2 1 (A. 17)
ED 4 Gkocosh2kohl (12/22 (A.17
where w2 = 2Gko2/p2.
Note 7: This semiempirical approach is based on field measurement in East Bay, Louisiana in water depth of 20 m
where the bottom sediments are composed of clays and silts. The instruments used included a wave staff, a pressure
sensor and a bottomemplaced accelerometer. The general characteristics of the data showed that the motions can
be adequately described in terms of the following harmonic functions:
p = po + plgAcos(krx wt) (A.18)
h2 = h20 + MAcos(krx wt + 0) (A.19)
where p = waveinduced bottom pressure, po = steadystate bottom pressure, A = surface wave amplitude, h2o
= depth of mud over which motion occurs, M = proportionality constant between the amplitudes of the mud wave
and the pressure wave; and 0 = phase angle between the crest of the pressure wave and the crest of the mud wave.
The mean rate of energy dissipation is then computed based on the average energy transmitted through the
sea/sediment interface per unit area over one wave cycle:
D 1 dh2 (A.20)
D 'T dt
where dh2 = an infinitesimal increase in the height of the interface. Using linear wave theory to interpret bottom
pressures in terms of surface wave height, and upon integration, the mean rate of energy dissipation due to mud
wave motion becomes:
CD P= ig wMsin a2 (A.21)
2D 2 coshkrhi
where O= 1800. Values of M and 4 can be obtained either from a theoretical approach or field measurement.
Note 8: The energy dissipation per unit volume in a linear viscoelastic medium over one cycle is given by:
wt2T6'
AE = rf dt = iroG" (A.22)
Wot 6'
where 6' = loss angle, r = shear stress, j = shear rate, yo = shear strain at steady state, G" = loss modulus.
Hence, the rate of energy dissipation (per unit volume) becomes:
S1 22 l 2 (A.23)
2v 2
where 1A' = dynamic viscosity.
By defining an average value of 7 by (Schreuder et al., 1986):
f J7d(wt+S')
<> 0 ___ 2y0 (A.24)
I T
O d(wt + ')
then,
E Z ] (A.25)
Cv= (2 'I<7>2) (A.25)
The same expression can also be obtained by considering the response of a viscoelastic material to a small
amplitude oscillatory shear. Given =ro0eilt, y=y7e'i(t"'), G*=7/7=Goe4't=G'iG" where G* = complex shear
modulus, G' = storage modulus and G" = loss modulus, the timeaveraged energy dissipation per unit volume over
a wave period, C,, reads:
T 2
i0 G"3'2 (A.26)
E, = Re(r)Re(i)dt = (A.26)
where Re denotes real part. By writing the complex viscosity as /*= '+i/", it can be shown that G"=wi/' and,
hence, the equivalence of the righthand sides of Eq. A.23 and Eq. A.26.
Assuming a uniform energy dissipation per unit volume across the depth of mud layer in analogy with that
of uniform energy dissipation in the water column due to breakinginduced turbulence, the mean rate of energy
dissipation per unit area can be written as:
00
ED = [ vdz (A.27)
h
Given h2=h2(x) as the equilibrium mud thickness,
ED = I ] (/'<'> 2h2) (A.28)
The above expression can then be evaluated by selecting an appropriate viscoelastic model, e.g. Voigt body, to
compute the relevant input parameters.
APPENDIX B. A SUMMARY OF SOME WAVEMUD MODELING STUDIES
Investigators) Water Mud Modeling Brief Description
Column Depth Theory
Gade (1957,
1958)
Mallard and
Dalrymple
(1977)
Dalrymple and
Liu (1978)
Dawson (1978)
Hsiao and
Shemdin (1980)
MacPherson
(1980)
Yamamoto
(1982, 1983);
Yamamoto and
Takahashi
(1985)
Mehta and Maa
(1986); Maa
and Mehta
(1987, 1989)
Inviscid
inviscid
Viscous
inviscid
Viscous" Shallow
water
Elastica Mud
thickness
unbounded
Viscous"
Elastic" Mud
thickness
unbounded
Inviscid Viscoelastica
(Voigt)
Inviscid Viscoelastic'
(Voigt)
Inviscid poroelasticb Multi
layered
mud
Viscous Viscoelastic'
(Voigt)
Multi
layered
mud
Linear Analytical approach for wave
attenuation; laboratory tests on
wave amplitude decay and
interfacial profiles, using a two
fluid system kerosinee overlying
watersugar solution).
Linear Analytical approach for soil
stresses, displacement and wave
kinematics. Soil inertia neglected.
No energy dissipation since
viscous property of mud not
considered.
Linear Analytical approach for wave
attenuation. Model valid for any
water depth and both deep and
shallow mud layers; boundary
layer approximation; explicit
solutions.
Linear Same as Mallard and Dalrymple
(1978), but soil inertia taken into
account.
Linear Analytical approach for wave
attenuation.
Linear Analytical approach for wave
attenuation.
Linear Based on nonlinear poroelastic
theory that includes nonlinear and
imperfect elasticity of soil
skeleton. Bed response nonlinear
and dynamically amplified;
primary dissipation mechanism
internal Coulomb friction between
grains. Arbitrary distribution of
continuously varying soil
properties with depth. Quasistatic
response more appropriate for
hard bed such as sands, while
dynamic response for soft beds
such as clays.
Linear Analytical approach with
laboratory tests on wave
attenuation and mud motion.
Investigators) Water Mud Modeling Brief Description
Column Depth Theory
Shibayama et
al. (1986)
Suhayda (1986)
Mei and Liu
(1987)
Tsuruya et al.
(1987)
Foda
(1989)
Viscous
Viscousa
Inviscid Viscoelastica
(Nonlinear)
Inviscid Viscoplastica
(Bingham,
thin layer)
Viscous Viscoplastic'
(Bingham)
Mud
thickness
unbounded
Shallow
water;
thin mud
layer
Multi
layered
mud
Inviscid Stratified Deep water;
Elastica mud
thickness
unbounded
Liu and Mei
(1989)
Viscous Viscoplastic"
(Bingham)
Shallow
water;
thin mud
layer
Linear Analytical approach using explicit
velocity field of Dalrymple and
Liu (1978) and laboratory tests on
mud mass transport and
attenuation rate.
Linear Analytical approach based on the
explicit results of MacPherson
(1980); assumptions of small
interfacial oscillations and infinite
mud thickness. Nonlinear
expressions relating shear strain to
shear stress and damping ratio.
Linear
and
Solitary
Analytical approach for wave
attenuation. Laminar constitutive
model and interfacial shear stress
< < mud yield stress. Predicts
continuous to intermittent mud
motion.
Linear Analytical approach by extending
Dalrymple and Liu's (1978)
viscousfluid model to Bingham
fluid. Laboratory tests on mud
mass transport and wave
attenuation. Equivalent viscosity
that takes into account the non
Newtonian mud rheology is
introduced in an approximate
fashion into the Newtonian
equations of motion and solved
iteratively.
Non Analytical approach for wave
linear attenuation. Wave energy lost to
highly dissipative sideband
oscillations, which in turn lose
energy due to viscous dissipation
whose action is enhanced by the
interaction of standing elastic
shear waves with viscous
boundary layer.
Solitary Analytical approach that includes
interfacial shear through boundary
layer approximation. Wave
dissipation due to bottom mud
shear layer and turbulent
interfacial stress. Predicts
continuous to intermittent mud
motion.
Investigators) Water Mud Modeling Brief Description
Column Depth Theory
Jiang and
Zhao (1989)
Chou (1989)
Shibayama et
al. (1989)
Sakakiyama and
Bijker (1989)
Viscous
Viscous
Viscousa Shallow
water
Viscous/
Viscoelastic/
Elastica
Viscous Viscoelastica
Viscous
Three
layered
mud
Multi
layered
mud
Viscousa
Solitary Analytical approach and laboratory
tests on wave attenuation based on
boundary layer approximation.
Linear Analytical approach for wave
attenuation and fluidization. A
generalized viscoelastic model
where the bed consists of two
layers with viscous/viscoelastic
properties overlying a halfplane
of viscoelastic/elastic muds, and a
viscoelastic boundary layer
approximation.
Linear Analytical approach and laboratory
tests on mud mass transport.
Linear Analytical approach based on
Dalrymple and Liu's (1978)
viscous fluid model with
laboratory tests on mud mass
transport.
Viscous' Shallow water Cnoidal
(1st
order)
Viscous Viscoelastic
and/or
viscoplastica
Power
Fluid'
Non
Newtoniana
Analytical approach for wave
attenuation based on boundary
layer approximation. A more
general model that includes those
of Jiang and Zhao (1989) for
solitary waves and Dalrymple and
Liu's (1978) for their explicit
solutions based on boundary layer
approximation under shallow
water waves.
Linear Extension of the model of
Shibayama et al. (1989) to include
viscoplastic bed when the shear
stress is greater than the Bingham
yield stress.
Shallow water
Linear
Analytical approach and laboratory
tests on wave attenuation and mud
fluidization. Power fluid (Sisko)
model obtained empirically.
Linear Numerical approach for wave
attenuation and mud mass
transport in which an empirical
rheologic mud model based on
multilinear approximation to
experimental stressstrain rate
curves is derived.
Viscous
Jiang et al.
(1990)
Shibayama et
al. (1990)
Feng (1992)
Isobe et al.
(1992)
inviscid
Inviscid
Investigators) Water Mud Modeling Brief Description
Column Depth Theory
Jiang (1993) Viscous Viscoelastic' Non Analytical approach and laboratory
linear tests on wave attenuation and mud
mass transport.
Cueva (1993) Viscous Viscoelastic Linear Analytical approach for wave
(Voigt) attenuation based on boundary
layer approximation
a Mud is assumed as a singlephase continuum.
b Twophased system consisting of soil skeleton and pore fluid in relative motion.