SOME OBSERVATIONS ON WATER WAVE ATTENUATION OVER NEARSHORE
UNDERWATER MUDBANKS AND MUD BERMS
Ashish J. Mehta
and
Feng Jiang
University of Florida
Gainesville, FL 32611
March, 1993
I.I "Jl. JiL`Ll AAA ~ ,. JI I
TABLE OF CONTENTS
LIST OF FIGURES .......... .................................. 3
LIST OF TABLES ............................................. 5
SYNOPSIS ................................................... 6
PART 1
MUDBANKS AND MUD BERMS AS MITIGATORS OF WAVE
IMPACTS ............................... ...
SUMMARY ............. ... ................... ....... 8
1.1 INTRODUCTION .................................... 8
1.2 BACKGROUND ..................................... 9
1.3 A SIMPLIFIED APPROACH TO BERM DESIGN ................ 15
1.4 CONCLUDING COMMENTS ............................ 22
PART 2
A RHEOLOGICAL MODEL FOR MUD ...................
. 23
SUMMARY ............ ................... ............
2.1 INTRODUCTION .................... ... ...........
2.2 CONSTITUTIVE BEHAVIOR ............................
2.3 EXPERIMENTAL EVIDENCE ............................
2.4 CONCLUDING COMMENTS ............ ...............
APPLICATION OF A VISCOELASTIC WAVEMUD
INTERACTION MODEL ............
.......... 31
SUMMARY ....... .............................. ...... 31
3.1 INTRODUCTION ............. ....................... 31
3.2 WAVEMUD INTERACTION ........................... 31
3.3 SOME MODEL APPLICATIONS .......................... 34
3.4 MOBILE BERM MUD RHEOLOGY ........................ 37
3.5 WAVE ATTENUATION OVER MOBILE BERM ................ 37
3.6 CONCLUDING COMMENTS ............................. 41
REFERENCES ..............................................44
PART 3
LIST OF FIGURES
Figure Page
1.1. Schematic drawing showing a natural coastal mudbank and a placed
underwater mud berm .................. ................... 9
1.2. Schematic sketch showing wave propagation over an underwater mud berm ... 10
1.3. Elevation sketch showing mudbank and its motion due to Stokes' drift ....... 11
1.4. Schematic profile of mudbank region off the coast of Kerala, India (after
Nair, 1988) ................ ......................... 11
1.5. Coastal site at Alleppey in Kerala, India where monsoonal, shoreparallel
mudbanks occur. The pier is 300 m long (after Mathew, 1992). ........... 12
1.6. Offshore and inshore wave spectra off Alleppey: (a) without mudbank, and
(b) with mudbank present (after Mathew, 1992). ................... 13
1.7. Construction site for the Mobile berm (after McLellan et al., 1990). ........ 13
1.8. 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, Hax: (a) Hma = 0.9 m, and (b) Hmax = 1.5 m (after McLellan
et al., 1990). ........................................... 14
1.9. 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 ...................................................18
1.10. Relationship between t and X representing the effects of wave energy
dissipation in mud on wave attenuation in shallow water. .............. 19
1.11. Relationship between r and x for determining the wave number corresponding
to Fig. 1.10 ........................................... 20
1.12. Effect of mud viscosity on relationships relevant to berm design: (a) wave
number versus water depth, (b) wave attenuation coefficient versus water
depth, and (c) bed shear stress amplitude versus water depth ............. 21
2.1. Mechanical analog of Eq. 2.4. It can be construed as a Voigt material with
additional elasticity G1, or as a fluid element constrained by additional
elasticity G2 ............ ............................. 25
2.2. Schematic plot of creepcompliance versus time for three applied stresses. ..... 26
2.3. Initial compliance versus applied stress for AK mud; data for four solids
volume fractions. ................... ..................... 27
2.4. Yield stress versus solids volume fraction for AK mud. . . . ... 27
2.5. Dependence of G1, G2 and p on stress amplitude and forcing frequency; data
for AK mud. ................... ....................... 28
2.6. Magnitude of viscosity Le versus frequency f for AK, OK and KI muds ...... 30
3. la. Wave attuation coefficient against frequency for two AK mud depths, 12 cm
and 18 cm. The water depth is 16 cm ........................... 35
3. lb. Comparison between measured and modelcalculated profiles of the amplitude
of horizontal acceleration in a watermud (AK) system in a flume. The water
depth was 16 cm, mud thickness 18 cm, water surface forcing wave amplitude
0.5 cm, wave frequency 1 Hz and = = 0.12. ......................... 35
3.2. Comparison between measured and modelsimulated inshore wave spectra off
Alleppey in Kerala, India: (a) fair weather condition (mudbank absent), and
(b) monsoon condition (mudbank present). ........................ 36
3.3. Experimentally determined variation of the viscoelastic parameters, G1, G2
and 4, with the forcing frequency, f, for MB mud. .................. 38
3.4. Experimentally determined variation of the coefficients in Eq. 2.6 with the
solids volume fraction, 4, for MB mud: (a) e versus 4, and (b) A versus ... 39
3.5. Comparison between measured and modelsimulated inshore wave spectra
at the Mobile berm site corresponding to two offshore wave conditions:
(a) Hmx = 0.9 m (see Fig. 1.8a), and (b) Hma = 1.5 m (see Fig. 1.8b)
Measured offshore spectra have been included for reference. ............. 40
3.6. Comparison between measured and modelsimulated inshore wave spectra at the
Mobile berm site. The mud rheology has been changed to a Voigt material
description ............................................. 42
LIST OF TABLES
Table Page
2.1. Coefficients of Eq. 2.6 for muds AK, OK and KI ...................... 29
3.1. Coefficients of Eq. 2.6 for MB mud .............................. 37
SOME OBSERVATIONS ON WATER WAVE ATTENUATION
OVER NEARSHORE UNDERWATER MUDBANKS AND MUD BERMS
SYNOPSIS
Naturally occurring underwater mudbanks are known to absorb water wave energy and
thereby attenuate waves that pass over. Energy reductions on the order of 30% to as much as
90% are not uncommon even in the absence of any measurable wave breaking. In recent years,
engineering efforts have been made to make use of this property of bottom mud by creating
underwater mud berms to mitigate wave impact in areas leeward of the berm. Thus, for
example, by appropriately placing finegrained dredged material from navigation channels in this
way the disposal site can be made to serve beneficially. In this report the dissipative property
of mud has been characterized via rheometry to simulate the degree of progressive water wave
damping over a mudbank or a mud berm. The report is divided into three parts as follows.
In Part 1 a shallow water wavemud interaction model is described and used to discuss
design parameters for a berm. Model application is given to determine the elevation of the berm
crest and the water column height above the crest in a given coastal environment. The model
considers water to be inviscid and mud to be a highly viscous fluid. The latter can be a
reasonable assumption in an environment where the top layer of mud that participates in the
dissipation process remains in a practically fluidized state. Such is for instance the case in Lake
Okeechobee, Florida where the bottom mud is composed of finegrained sediment with 40% (by
weight) organic matter. On the other hand, in more typical coastal situations the elastic
properties of mud must be included in the theological description. Since the Voigt viscoelastic
description, the most commonly used constitutive model for mud rheology in wavemud
interaction studies, is not fully applicable over the range of natural forcing frequencies and mud
densities, a new model is described in Part 2. The basis of this model is a series of experiments
carried out using a controlledstress rheometer, CarriMed CSL, at the Waterways Experiment
Station, Vicksburg, Mississippi. The Voigt model is a special case of the developed model at
comparatively high forcing frequencies.
In Part 3 a finite amplitude (nonlinear) wavemud interaction model that is not restricted
to the shallow water condition is described. Mud rheology is based on the constitutive model
described in Part 2. Water is taken as a viscid fluid in this case, and provision is made to
include a turbulent diffusion coefficient for the water column. This model has been previously
tested against laboratory flume data, and is used here to calculate wave attenuation at certain
highly mobile monsoonal mud banks off the southwest coast of India. There the damping is often
so significant that offshore storm waves practically vanish by the time of their arrival at the
shoreline.
The above model is finally used to calculate wave attenuation over a nonsacrificial mud
berm (also called the Mobile Outer Mound) designed by the Army Corps of Engineers in the
Gulf of Mexico, off Dauphin Island in Alabama. Finegrained material for the berm was derived
from dredging the ship navigation channel into Mobile Bay. The berm has been effective in
reducing the wave energy in the area sheltered by the berm. In two cases considered for
application, 29% and 46% energy reductions were measured. Considering the nature of the wave
field and the water depth over the berm crest at the site, the high degree of wave damping is
believed to be mainly due to wave energy absorption over the berm. Using the measured wave
spectrum offshore of the berm and the native mud rheology, the model is shown to simulate, to
a reasonable degree, the wave spectrum inshore of the berm, where a measured spectrum was
available for comparison purposes.
The above exercise demonstrates the importance of appropriately characterizing mud
rheology for calculating the degree to which wave damping will occur across a mudbank or a
mud berm. The model can be used as a tool in determining the crest elevation of the berm and
its offshore placement distance for a given design reduction in wave energy. It can also be used
to assess potential wave impacts where a preexisting berm or a bank is removed naturally or
otherwise.
Support from project DACW3993M1483 for this work is acknowledged.
PART 1
MUDBANKS AND MUD BERMS AS MITIGATORS OF WAVE IMPACTS
SUMMARY: The role of nearshore bottom mud in absorbing water wave energy, hence in
mitigating potentially adverse wave impacts in leeward areas, is examined in the natural
mudbank setting and as a mechanism that must be understood for the design of protective
underwater berms. A shallow water wavemud interaction model that assumes mud to be a
highly viscous fluid is described and used to consider the design of a stable mud berm.
Specifically, the model is used to determine the berm crest elevation and the water depth in a
given nearshore environment and mud properties.
1.1 INTRODUCTION
The capacity of naturally occurring bottom mud to absorb wave energy in coastal areas
has been known for years, and in recent times this property has been recognized as a potential
means for protecting beaches that are susceptible to erosion under wave attack. In the schematic
drawing of Fig. 1.1 a natural mudbank as well as a placed underwater berm are shown. As an
example of the former, prominent mudbanks occur on the eastern margin of the Louisiana
chenier plain where episodic waves are measurably damped as they pass shoreward over the
banks. Their length, 1M, varies from 1 to 5 km, width, wM, from 0.5 to 3 km and thickness
from 0.2 to 1.5 m, in water depths on the order of 1 to 2 m above the mud surface. The median
particle size (3 to 5 pm) is in the fine silt range (Wells, 1983; Wells and Kemp, 1986).
As an example of a nonsacrificial, engineered berm, off Dauphin Island in Alabama, an
underwater mound composed of sediment dredged from the Mobile Ship Channel within Mobile
Bay was deposited by the Army Corps of Engineers in 1988. This berm, also called the Mobile
Outer Mound, is a part of a threeberm demonstration project to evaluate their stability and
potential usefulness for reducing wave damage to the shore (Hands, 1990). The design
dimensions of the berm placement "corridor" were: 2,750 m length and 300 m width. Placement
of 1.3x107 m3 of dredged material has created a berm of very approximately 6 m height and
with 6.5 m mean depth of water above the berm crest (McLellan et al., 1990). The placed
material is highly graded, with a finegrained median size on the order on 15 pm (McLellan,
personal communication; Dredging Research Technical Notes, 1992).
The degree to which such coastal features, natural or otherwise, attenuate incoming wave
energy depends on their dimensions, the composition of the sediment and the incoming wave
characteristics. Thus a predictive method, for instance for berm design, must be inclusive of the
essential physical modes of interaction between the flow and the dissipative bottom. With regard
to berm design the two most important parameters are the crest elevation and the depth of water
above the crest, since they measurably influence the degree of wave attenuation, hence wave
impact in sheltered areas. Relevant considerations for berm design are briefly described in what
follows.
Sea
WM 1 /
T
.:' : ,. ,' ..'.. : : .':. ; ,':.:..
Figure 1.1. Schematic drawing showing a natural coastal mudbank and a placed underwater mud
berm. Plan dimensions I and w are subscripted by M and B, respectively, for the mudbank and
the berm.
1.2 BACKGROUND
The physical setting relative to the mudbank or the underwater berm is conceptually
shown in Fig. 1.2. Seawater (density p, dynamic viscosity p) surrounds the mud berm, which
is in general a viscoelastic material (density p ) possessing both an elastic component
characterized by the shear modulus, G [O(101104) Pa], and viscosity trM [O(101104) Pa.s].
Most muds are pseudoplastics in terms of the relation between the shear stress and the rate of
strain, so that they exhibit creep even at very low applied stresses.
When subjected to wave action, bottom mud responds by oscillating predominantly at the
forcing wave frequency, although as a result of high viscosity the oscillations (particulate orbits)
attenuate much more rapidly with depth within mud than in the water column above. The high
rate of dissipation in turn causes the wave height to decrease rapidly with onshore distance.
Thus, given the wave amplitude ao at x = 0 at the seaward edge of the berm crest, the
amplitude at any distance x, i.e. ax = aoexp(kix), depends on the wave attenuation coefficient,
ki. If the bottom were rigid, ki would be O(10") ml, whereas for mud much higher values,
O(104103) m1, are common (Jiang and Mehta, 1992). The result is much reduced wave
activity in the area leeward of the berm in the latter case compared with areas where the bottom
is for instance sandy. On the other hand the wave celerity, C, is higher where mud is the
substrate, because the available depth for wave propagation is effectively greater than the water
depth, h. Consequently, a parameter that requires specification is the depth hM' in Fig. 1.2
Figure 1.2. Schematic sketch showing wave propagation over an underwater mud berm.
corresponding to the thickness of the mud layer that effectively participates in the dissipative
process through mud motion. Recent studies by, e.g. Feng et al. (1992), suggest that hM'
depends on, among other factors, the rate of wave energy dissipation, i.e. that the two can be
related empirically.
Mud oscillation primarily occurs as a result of waveinduced pressure work within the
body of the material, while the effect of shear stress is more important at the mud surface where
it can cause particulate resuspension. Thus, under continued wave action the equilibrium water
depth above the crest is that depth at which the waveinduced stress (amplitude), rbm, is equal
to the erosion shear strength, r7, of mud. This being the case, in many naturally occurring
environments mud oscillations under typical fair weather wave conditions occur without much
particulate resuspension and associated turbidity (Jiang and Mehta, 1992). In turn this condition,
i.e. rbm = Ts, can be used to design the berm crest elevation, so that the berm can fulfill its role
as a wave attenuator without generating excessive turbidity and selfdissipating in the process
through transport of the eroded sediment.
For a given crest elevation the slope from crest to toe is determined by the thixotropic
yield strength, which for design purposes has been approximated by the upper Bingham yield
strength, ry, of the pseudoplastic material (Migniot, 1968). However, the stability of the berm
crest is not assured solely through the aforesaid criterion for erosion since, due to open ("non
closed") particle orbits (Fig. 1.3) arising from nonlinear wave effects, the residual velocity, uL
(Stokes' drift) can cause the mud mass to be transported landward. The impetus for this motion
is the net waveinduced thrust that occurs in the mud due to the rapid, wave attenuation with
distance. Thus, hypothetically starting at time t with a mudwater interfacial profile that can be
theoretically shown to be nearexponential in form (Jiang, 1993), the depthaveraged value of
uL will become nil everywhere at some instant te, when an equilibrium interfacial profile is
established. Such a condition will occur due to a balance between the waveinduced thrust and
the adverse hydrostatic gradient in the presence of a sloping bottom is shown in Fig. 1.3.
Profiles such as these are seasonally established for instance along the southwest coast of India
off Kerala, where the southwesterly monsoon waves push the mudbanks from the inner shelf in
about 20 m depth to a position close to the shoreline. Figure 1.4 is a schematic profile of the
Profile)
Figure 1.3. Elevation sketch showing mudbank and its motion due to Stokes' drift.
Peripheral Wave
Damping Zone High Water Line
Calm Water Area
Intense Wave Activity Mud Bank Zone
Mud in Suspension o
Figure 1.4. Schematic profile of mudbank region off the coast of Kerala, India (after Nair,
1988).
mudbank at its nearshore location (Nair, 1988). Its position has been stabilized against the
comparatively steep nearshore bottom.
Mass transport due to Stokes' drift is important in situations such as during the Indian
monsoon when sustained and significant wave action may occur for weeks or months at a
stretch. Furthermore, the occurrence of a gravity slide into deeper waters at the cessation of high
wave action towards the end of the monsoon causes the situation to be reinitialized with respect
to mass transport at onset of the following monsoon. Field evidence elsewhere, e.g. off Surinam
(Wells, 1983), indicates that unlike the Kerala mudbanks which are clearly quite transitory,
mudbanks typically tend to remain practically stationary because of small bottom slopes, mild
and/or constant wave action, or otherwise. Consequently their net motion is not a matter of
general interest, and need not be considered in simple calculations for stable berm design.
Some examples of a drastic reduction in the wave energy due to wave passage over
mudbanks and berms are worth a note. Figure 1.5 shows the region offshore of the town of
Alleppey in Kerala where wave spectra were obtained at two sites (Mathew, 1992). 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. During monsoon the shoreward edge
Figure 1.5. Coastal site at Alleppey in Kerala, India where monsoonal, shoreparallel mudbanks
occur. The pier is 300 m long (after Mathew, 1992).
of the bank was shoreward of the inshore site, while the seaward edge of the bank was
shoreward of the offshore site. Thus the bottom at the offshore site was consistently devoid of
mud. Figures 1.6a,b respectively show examples of wave spectra in the absence and in the
presence of the mudbank at the inshore site. In fair weather the energy reduction was negligible.
(Note the slight phase shift of the inshore spectrum relative to offshore. This shift may be an
artifact of the data analysis; however, if a physical basis was present it remains uncertain.) In
the monsoon case the energy reduction 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, during monsoon the mudbanks in Kerala have served as safe, open coast, havens
for small fishing boats that would otherwise require sheltered harbors (Nair, 1988).
The dredged material placement corridor for the Mobile berm is shown in Fig. 1.7.
Examples of offshore/inshore wave spectra are given in Figs. 1.8a,b under two different offshore
wave conditions. Wave energy reductions were significant, 29% and 46% respectively, although
not nearly as dramatic as in Kerala because, 1) the finer and more clayey Kerala mud is more
dissipative, and 2) the berm is quite narrow compared with the mudbank. Using a 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 depth of water, on the
order of 6.5 m over the berm, the main cause of damping is energy absorption by the deposited
mud. Interestingly enough, diver observations at the site have suggested the occurrence of
surface waveforced interfacial mud waves propagating along the compliant crest (McLellan,
0.2 0.4
FREQUENCY,f (Hz)
FREQUENCY,f (Hz)
Figure 1.6. Offshore and inshore wave spectra off Alleppey: (a) without mudbank, and (b) with
mudbank present (after Mathew, 1992).
\Sand Island
SMobile
/ Point
Inshore i
Offshore /
I/,
Figure 1.7. Construction site (corridor for dredged material placement) for the Mobile berm
(after McLellan et al., 1990).
0.
0.
0.
U I I I
(a) 7/27/87 
8 Mudbank Absent 
Mathew (1992)
6
4
SOffshore
2 ~Inshore
0
ofi
0.0
0.25 1 1 1
N (a)
E McLellan et al. (1990)
> 0.20 Hmax 0.9 m
I
Z
S0.15 Offshore
C( 0.10 
LU
z I
w 0.05 
W Inshore
0
0 0.1 0.2 0.3 0.4 0.5
FREQUENCY (Hz)
S1.0 I I I
(b) McLellan et al. (1990)
0.8 Hmax = 1.5 m
0.8 
Z 0.6 
0
S0.4
w r
w 0.2
0 0.1 0.2 0.3 0.4 0.5
FREQUENCY (Hz)
Figure 1.8. 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, HIax: (a) Hax =
0.9 m, and (b) Hmax = 1.5 m (after McLellan et al., 1990).
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 observed
stability of the berm points to the fact that wave action becomes sufficiently weak over the berm
to prevent significant waveinduced scour and degradation of the berm.
1.3 A SIMPLIFIED APPROACH TO BERM DESIGN
To focus on the berm elevation and the water depth, we will assume the berm to be
sufficiently long as to enable us to ignore the end effects, thus considering the structure to be
endless in the alongshore direction. This assumption requires the berm to be substantially long
in relation to the length of the wave. End effects can also be prominent in the crossshore
direction, especially if the side slope is steep and the crest elevation high enough to cause an
abrupt depth change at the seaward edge. Furthermore, a change in the sediment composition
between the ambient bottom and the berm may also measurably influence the wave field. The
mechanics of propagation of the wave over the berm would thus become an initial (spatially)
value problem, the initial condition being defined at the toe of the berm. For typical natural
mudbanks the crest elevations, hence side slopes, are not high however, and in fact the only real
change encountered sometimes is that in the bottom composition. Even at berms constructed by
dredged material placement, side slopes are not always excessively steep; at the Mobile site the
slopes range from 1:24 to 1:130. Thus, for simplicity the effects of changing depth and bottom
composition on the dissipation process over the berm can be assumed to be negligible.
Consider wave propagation in a single (x) direction in water of depth h, over mud of
thickness hM (= hM'). Harmonic solution is sought for the wave speed and other waveinduced
quantities. As for the theological constitutive properties of mud, viscoelastic models of different
levels of complexity have been considered (e.g. Chou, 1989; Jiang, 1993). Recent studies on
wavemud interaction (e.g. by Feng et al., 1992) however suggest that wave pressure work can
cause the otherwise undisturbed mud to weaken through thixotropic yield leading to plastic
behavior and loss of effective stress. This process of fluidization is reflected in a drastic
reduction in the modulus of elasticity (G) with the result that the weakened mud can be
approximated as a highly viscous fluid. Here we will assume mud to be as such and consider
water to be inviscid for simplicity. Note that in Part 2 the elastic component has been added for
a more accurate description of mud rheology, and the resulting wavemud interaction model has
been described in Part 3. For the present we will also invoke the shallow water assumption and
consider the wave propagation problem to be linear. Note that these two assumptions have been
relaxed in Part 3 to account for the effects of finite amplitude (nonlinear) waves in intermediate
(i.e. nonshallow) water depths.
As a result of the shallow water assumption, only horizontal motion is considered in the
governing equations of motion and continuity, which can be written as (Jiang, 1993):
Ul r = o (1.1)
g=0
at ax
(_ hau! = 0 (1.2)
t ax
for the water layer (characterized here by subscript 1), and
u2 g 2 (1,) 1 2u2 (1.3)
+ 7yg + (l) = v 2 2
at ax ax az2
f hi U2dz O+ 2 0 (1.4)
Jo Ox at
for the mud layer subscriptt 2). Here x,z are the horizontal and the vertical axes located at the
rigid bottom, il1(x,t) and (2(x,t) are functions of x and time t, ul(x,t) and u2(x,t) are the wave
induced velocities, is the normalized density jump, hi is the elevation of the watermud
interface, g is the gravitational acceleration, and v is the kinematic viscosity of mud. Thus, y
= (p2P)/P2, hi = h2 + 712 and v2 = 2/IP2, where /2 is the dynamic viscosity.
Based on the considerations that the fluid domain bounded between z = 0, h1 + h2 is
infinite in extent in the x direction, mud is viscid and surface loading is a linear harmonic
wave, the boundary conditions can be expressed as:
il1(0,t) = a0cosot (1.5)
uI(oo,t), u2(co,z,t), 41(oo,t), 02(oo,t) = 0 (1.6)
u2(x,0,t) = 0 (1.7)
au2(x,h2,t) 0 (1.8)
Oz
where ao is the surface wave amplitude at x = 0. Equation 1.5 specifies a progressive, simple
harmonic wave of angular frequency a at the water surface. Equation 1.6 requires that wave
motion must cease at an infinite distance by virtue of viscous energy dissipation in the mud
layer. The noslip bottom kinematic boundary condition is stated in Equation 1.7. Equation 1.8
represents the condition that due to the assumption of an inviscid upper layer fluid (water), there
can be no shear stress at the watermud interface. Note that as a result of rotationality in the
lower layer and the shallow water condition, u2 varies with z, but not ul.
Harmonic solutions for 71, U1, 72 and u2 are presented by Jiang (1993). The validity of
the results has been demonstrated previously in laboratory experiments by Gade (1958). The
results have also been applied to measurements obtained at a site in the littoral margin of Lake
Okeechobee, Florida, where about a third of the bottom is composed of a layer of mud (having
a maximum thickness of 0.8 m) of peaty origin (Jiang and Mehta, 1992). Wave damping is often
significant; during a field deployment it was estimated that the wave heights were about one
quarter the value that would have occurred if the bottom had been rigid. Data at the nearshallow
water site consisted of surface wave, water velocity and mud acceleration spectra. The water
depth was 1.43 m and mud thickness 0.55 m. Using the wave spectrum as input, the velocity
and acceleration spectra were predicted and compared with data. The highly organic (40% by
weight) mud was fluidlike, with a comparatively low viscosity of 5 Pa* s.
In Figs. 1.9a,b,c an example of model application is shown (Jiang, 1993). The input
wave spectrum of Fig. 1.9a has been used to predict: 1) the horizontal water velocity spectrum
(0.87 m below the water surface) in Fig. 1.9b, and 2) the horizontal mud acceleration spectrum
(0.20 m below the mud surface) in Fig. 1.9c. Observe that while the modal forcing frequency
was 0.42 Hz, a forced long wave occurred at 0.03 Hz. The significance of this wavewave
interaction has been discussed by Jiang and Mehta (1992). The model is able to reproduce
important features of the velocity and acceleration spectra. On the other hand, the observed
discrepancies between the measurements and the simulations is believed to be mainly due to
limitations in the model, especially as a result of the assumption of a viscid (nonelastic) mud
(Jiang, 1993). Some difficulty in collecting the data may have been a contributing factor as well
(Jiang and Mehta, 1992).
Recognizing the model assumptions, we will proceed with graphically presenting the
analytic solutions for the wave attenuation coefficient and the wave number. In Fig. 1.10, the
dimensionless wave attenuation coefficient, E = kl/[a/(gh)1/2], is plotted as a function of the
dimensionless mud depth, x = hM/(2MpM/pMa)1/2, for hM/h ranging from 0.1 to 1. This solution
is valid only for y = (pmp)IM p = 0.15, which is however a typical value. Note that in this
representation, subscript 1 for water has been removed for convenience, and subscript 2 for mud
has been replaced by M, in conformity with Fig. 1.2. It can be shown that (2/MIpMa)112 is
representative of the thickness of the wave boundary layer in mud. For a given water depth and
angular wave frequency a, attenuation is observed to be maximum for all hM/h when X is equal,
or close, to unity. Therefore, this condition of maximum ki means that the rate of wave energy
dissipation in the mud is maximum when the wave boundary layer attains about the same
thickness as mud. Two noteworthy limits must be mentioned. One is X = 0, which corresponds
to a mud of infinitely high viscosity, i.e. a rigid bottom. In this case there is no participation by
mud in the dissipation process, and since water is assumed to be inviscid there is no damping.
The other limit is approached with increasing X, i.e. decreasing mud viscosity. In this case
damping also approaches zero as suggested by the plots.
U)5
50' 50
2E I II /
(a) Lake Okeechobee, Florida 
> 40
!
,u 30 
O 20
W
z
w 10
0
2o iI Simulation
v (t 
j 120 '
.>
> measurement
.W 60
0.40
6T (c)
E
a 0.30
zo
cc 0.20
W

0 /01
iQ / l"\
0.00
0.0 0.2 0.4 0.6 0.8 1.0
FREQUENCY (Hz)
Figure 1.9. 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.
)0.7
c
t .0.6
S0.08 ".=0.15
0.40.04
0.30.00
0.04 0.2
0 1 2 3 4 5
X = hM/(2pM/PM),'2
Figure 1.10. Relationship between and X representing the effects of wave energy dissipation
in mud on wave attenuation in shallow water.
In Fig. 1.11 the companion plot for determining the wave number, k, is given. The
dimensionless wave number = k/[a/(gh)1/2] is shown as a function of X. At x = 0, = 1
corresponds to shallow water wave motion in depth h, whereas with increasing X participation
by mud increases, and in the limit the problem becomes one of inviscid water wave propagation
in depth h+hM. Note that the solution for the maximum (at the seaward edge of the berm crest)
water velocity amplitude is um = a0gk/a, where g is acceleration due to gravity. Thus the shear
stress amplitude at the mud surface, rbm = tUm2, where a incorporates water density and a drag
term which depends on the roughness of the mud surface as well as the amplitude of the water
particle just above the mud surface (Dyer, 1986). We will make use of these solutions to address
a simple, hypothetical problem of berm design.
We are required to place a 3 m high (hM'= hM) finegrained, dredged material (density
1,180 kg/m3) mound structure in water such that it remains stable. At the same time, economics
of dredged material transport dictates that the structure be placed in the shallowest possible
depth to minimize the distance between the shoreline and the structure. Other relevant
parameters are: wave frequency a = 0.2r rad/s, forcing wave amplitude ao = 0.75 m, erosion
shear stress rs = 1 Pa and a = 1.03 Pa.s2/m2 (for a hydraulically smooth turbulent flow). We
will examine the effect of mud viscosity on the problem by selecting two values of the
mud cinematic viscosity, vM = tM/PM = 103 and 101 m2/s. We will first find the water depth
h, then calculate wave attenuation, assuming the berm to be 300 m wide (wB).
For this case x = 53 and 5.3, respectively, for the low and high viscosity values.
Figures 1.11 and 1.10, respectively, can then be used to yield the dependence of the wave
number, k, the wave attenuation coefficient, ki, and the bed shear stress amplitude at the berm
i.P 0.8 7=0.15
0.0.8
0.9
1.0
o.7 1  I  I  I I  I  I  I  I  I 
0 1 2 3 4
X = hM/(2p.M/pMa)12
Figure 1.11. Relationship between and X for determining the wave number corresponding to
Fig. 1.10.
crest, Tbm, on the water depth, h. In Figs. 1.12a,b,c these relationships are shown graphically
for depth h ranging from 1 to 8 m. Note that the model cannot be used for greater depths due
to the shallow water assumption. The wave number is seen to be comparatively insensitive to
viscosity, since it depends primarily on the inertia and pressure forces in the water column. The
same is observed in the case of the bed shear stress. (Note also that the magnitude of the bed
shear stress is an artifact of its chosen dependence on the water velocity in the inviscid water
column under the given assumptions. This dependence is characterized by the coefficient a.) On
the other hand wave damping is seen to be highly sensitive to the viscosity, and emphasizes the
need to appropriately consider the composition and the constitutive behavior of the substrate in
the problem.
Given the erosion shear stress r. = 1 Pa, the minimum water depth required for a stable
berm would be 3.2 m (Fig. 1.12c), selecting the higher of the two viscosities as realistic. The
site where this depth occurs will be the minimum distance from the shoreline where the berm
must be placed for it to be nonsacrificial. Since for the depth of 3.2 m the wave attenuation
coefficient ki = 0.002 1/m, the 1.5 m high wave will be reduced to 0.82 m, which corresponds
to a 70% reduction in the wave energy.
20 I I I I I
vM (m2/s)
16 0.1
 0.001
12
08
04
^ ^ I I I I I I I I I
u.uu
0.
0.
0.
0.
0.005.. 1 i
0.004
0.003 
0.002 < 
0.0011 'I
S Jnnnr
E
kj2
U)
I)
C1
Cl)
0 2 4 6
WATER DEPTH, h (m)
8 10
Figure 1.12. Effect of mud viscosity on relationships relevant to berm design: (a) wave number
versus water depth, (b) wave attenuation coefficient versus water depth, and (c) bed shear stress
amplitude versus water depth.
(c)
I I I l l i l
1.4 CONCLUDING COMMENTS
Information on the berm crest elevation and the minimum water depth for a stable berm
must be supplemented by other design parameters for engineering implementation of berm
design. Among these the berm slope is perhaps most important. In that context the relationship
of Migniot (1968), tana' = fry, where a' is the slope (angle), is worth a note. From laboratory
experiments the empirical constant 1 was found to vary from 0.007 to 0.025 when the yield
stress, ry, was measured in Pa. If for the sake of argument we assume 7y = T, = 1 Pa, the
corresponding range of slope would be 1:143 to 1:40, which is realistic, when compared with,
for instance, 1:130 to 1:24 at the Mobile berm.
PART 2
A RHEOLOGICAL MODEL FOR MUD
SUMMARY: A three parameter theological constitutive model is described to account for the
viscoelastic properties of mud. The associated elastic shear moduli and viscosity have been
measured in a controlledstress rheometer. Within specified ranges of the applied shear stress
and the forcing frequency the model is shown to apply to three muds with differing physico
chemical properties.
2.1 INTRODUCTION
Accurate prediction of wave energy dissipation within mud requires a constitutive
equation that must account for the viscoelastic properties of mud. Thus, consider the equation
of motion for a nonNewtonian medium in the xdirection,
Du 1 ap a2u 2u (2.1)
Dt p ax e ax2 z2
in which D denotes a total derivative, u is the horizontal velocity, z is the vertical coordinate,
t is the time, p is the pressure, p is the mud density and Ve is defined as the equivalent kinematic
viscosity. The corresponding dynamic viscosity, /lt = pVe, depends on the choice of the
theological description, and Eq. 2.1 has been shown to be theoretically valid for viscoelastic
materials subject to oscillatory loading (MacPherson, 1980). The application of this equation for
modeling wave attenuation is considered in Part 3. Here an expression for et is developed
assuming mud to be a continuum, and its coefficients examined through controlledstress
rheometry involving three muds of different physicochemical properties.
One mud (AK) was a 50/50 (by weight) mixture of an attapulgite and a kaolinite in
aqueous solution with a median size of about 1 /m. Its granular density, p,, was 2,650 kg/m3.
The second (OK), in native water, was obtained from the shallow bottom of Lake Okeechobee,
in which kaolinite was the dominant clay mineral together with some smectite and sepiolite, with
a high organic content (40% weight loss on ignition). The median grain size was about 10 1tm
and granular density 2,140 kg/m The third mud (KI), also in native water, was obtained off
the Alleppey in Kerala, India. It was composed of kaolinite and smectite in about equal
proportions together with small amounts of illite and gibbsite. The median size was 2 jtm and
granular density 2,650 kg/m3. The samples were tested in a CarriMed CSL rheometer with a
cylindrical measuring geometry (bob) at the Waterways Experiment Station.
2.2 CONSTITUTIVE BEHAVIOR
The general constitutive equation for viscoelasticity is,
r + oira + o2r + ... = 7 + 215 + 2 + ... (2.2)
in which r is the shear stress, is the corresponding strain and the dot signifies derivative with
respect to time. Thus is the timerate of strain, the rate of strain rate and so on. (Note that
unlike in Parts 1 and 2, subscripts 1 and 2 here do not refer to water and mud respectively; both
are for mud.) The coefficients a,# depend on material properties and must be determined
experimentally. In a linear viscoelastic material these coefficients are independent of stress and
strain and their timederivatives. If further all can's and #'s are zero with the exception of #1, Eq.
2.1 reduces to one describing a Newtonian fluid (#f = /, the dynamic viscosity), i.e. r = tj'.
Similarly, given #o = G, the shear modulus of elasticity, r = Gy is the elastic response of a
Hookean solid. Thus a viscoelastic Voigt material is described by r = Gy + j/', with a
characteristic strain retardation time constant, G//x, associated with material response to applied
stress. If oa and #1 were the only nonzero parameters the Maxwell fluid element, r +
(O/G) = /s', would result. The Voigt and the Maxwell elements, two fundamental forms of
viscoelastic descriptions, have been considered in describing mud rheology and shown to be
applicable to selected situations involving wavemud interactions, but are generally found to be
deficient (e.g. Chou, 1989).
Since real materials are characterized by spectra of retardation time constants, the
complexity of material response to stress requires that the constitutive equation be tested against
data to validate the choice of the model. By way of such an approach a model that is applicable
to the selected material within specified parametric bounds can be found, even though uniqueness
of the model cannot be established. Physically the requirement is to combine the Voigt solid and
Maxwell fluid responses, recognizing that comparatively dense muds retain residual stresses
when strained (Maa and Mehta, 1988). Mathematically these conditions can be shown to lead
to a model that would, in its simplest form, include the 0th and the 1st timederivatives of r and
y (Malvern, 1969). Thus
7 + a1t = P?' + f1ij (2.3)
This model has been used in describing the consolidation behavior of soils (Keedwell, 1984).
Figure 2.1 shows the springdamper mechanical analog of Eq. 2.3 in which setting the shear
modulus G1,oo reduces the analog to the Voigt model, while G2 = 0 yields the Maxwell
model. Equation 2.3 can be written in terms of t4, G1 and G2 as
+ + G (2.4)
G +G2 G1+G2 G+G
and the viscosity, /e, expressed as
G,
G2
Figure 2.1. Mechanical analog of Eq. 2.4. It can be construed as a Voigt material with
additional elasticity G1, or as a fluid element constrained by additional elasticity G2.
Ile (1+T)2+S2
= 1 l+i{R(1 +T) +S}] (2.5)
where R = G2/zUr and S1 = Gl//,a are recognized as dimensionless retardation time constants,
T = G2/G1, a = angular frequency and i = \rT. The coefficients R, S and T and hence 4,
G1 and G2 must be evaluated through a combination of static and dynamic rheometric tests. In
the former (creep test) a constant stress, r0, is applied instantaneously to the sample for a
duration t, and then removed instantaneously. The induced strain is timetracked. In the latter
test a sinusoidally oscillating stress of amplitude ro is applied and the corresponding strain
response tracked with time.
A considerable advantage of controlledstress rheometry over one in which the strain is
controlled, i.e. inputted, is that in the former case the soil fabric is not necessarily disturbed
irreversibly as always occurs when the sample is strain controlled. In fact, controlledstress tests
provide a means to examine the true yield behavior under conditions when thixotropic
deformation does occur. In Fig. 2.2, the creepcompliance, J = y/ro, is plotted against time
schematically for three independent tests in which 7r is increased sequentially. The first test
shows viscoelastic strain recovery. The second test corresponds to incipient thixotropic yield,
hence To = Ty, the yield stress equal to the upper limit of 7T for which measurable elastic and
retarded elastic responses associated with a continuous, threedimensional network structure are
reached (James et al., 1988). This yield stress is in general different from the upper Bingham
yield stress (also denoted 7y in Section 1.4). Finally, the third test shows strain behavior
corresponding to yield. As suggested by the curves the variation of the initial, instantaneous
compliance, Jo, a measure of elastic response, with the applied stress is related to the state of
the material.
The reader is referred to Jiang (1993) for creep and oscillatory test data using the three
muds and the method for calculation of 1/, G1 and G2 as functions of ro and the forcing
frequency, f (= a/2r), for selected values of the mud density [in terms of the solids volume
fraction 4 = (ppf)/(Pspf), where pf is fluid (water) density]. Illustrative results and their
significance are briefly described in the next section.
J = Y/To
o > Ty
to <
T= 0
t> tc
Figure 2.2. Schematic plot of creepcompliance versus time for three applied stresses.
2.3 EXPERIMENTAL EVIDENCE
Figure 2.3 is a plot of initial compliance, Jo, versus ro for AK. The curve for each
volume fraction 4 can be approximately divided by the dashed line representing the yield
condition. For stresses lower than ry, J0 generally increases rather gradually with o0, with the
rate of increase decreasing with increasing 4. In fact, with the possible exception of the curve
for f = 0.03 the dependence of J0 on ro is weak. This behavior in turn implies a practically
linear viscoelastic response. Once yield commences Jo increases rapidly with ro. Comparing the
curves for increasing q, a dramatic corresponding rise in 7y becomes apparent as shown in
Fig. 2.4. This rise is associated with the development of a spacefilling network structure as
4 exceeds a "critical" value, 0f. The data suggest 4q to be approximately 0.05, which is
comparable with 4, for other muds (James et al., 1988). This value in turn provides a possible
explanation for the Jo versus ro curve for f = 0.03 in Fig. 2.4, which shows no clearly
identifiable value of ry, since spacefilling was evidently incomplete in that case.
For mud of given volume fraction the behavior of Eq. 2.4, hence Ae (Eq. 2.5), can be
examined in terms of the dependence of 1, G1 and G2 on o0(< y) and frequencies that
encompass those relevant to the natural wavemud environment. In Fig. 2.5 for AK (4 = 0.12)
the shear moduli and the viscosity show a relatively weak dependence on ro, thus implying a
practically linear response over a rather wide frequency band (0.02 to 4.5 Hz). The lines can
be expressed generally as
I I
AK Mud

= Ty
I I
' I. 'I
0 0.03
0.06
A 0.12
A 0.19
I I
I I
101 100
STRESS, to (Pa)
Figure 2.3. Initial compliance
fractions.
w
I
CD
0
j
iQ
^J
UJ
>*
versus applied stress for AK mud; data for four solids volume
VOLUME FRACTION, 4
Figure 2.4. Yield stress versus solids volume fraction for AK mud.
I I .
101
102
y (Pa) 
0.03
0.6
10
30
103 
104 
UI I I I I I I I
102
I
105
104
10
0 10 0 0 0 0 0
M 103
o T (Pa)
3 A 0.3
C 102 a 0.6
0 AKMud
S100
101
o 10
106
a
1025 r
:: 104 0
10
0 
102_
102 101 100 101
FREQUENCY, f (Hz)
Figure 2.5. Dependence of G1, G2 and / on stress amplitude and forcing frequency; data for AK
mud.
G1 or 2 = exp().fA (2.6)
for which values of the parameters e and A are given in Table 2.1. The frequency dependence
of G1, G2 and ts is related to the corresponding relation between the rate of energy dissipation
and frequency. This relation is believed to arise due to the thermodynamic state of the
dissipation process, which changes from isothermal at low frequencies to adiabatic at high rates
of structural vibrations (Krizek, 1971; Schreuder et al., 1986).
of structural vibrations (Krizek, 1971; Schreuder et al., 1986).
S I I I 1111 I IIII I I III I I I I I III
Table 2.1. Coefficients of Eq. 2.6 for muds AK, OK and KI
Mud GI(Pa) G2(Pa) /(Pa. s)
E A e A E A
AK 0.12 8.049 0.114 2.604 0.490 8.222 0.490
OK 0.11 5.548 0.127 0.318 0.687 5.290 0.687
KI 0.12 9.160 0.257 3.843 0.405 9.292 0.405
For a comparative purpose values of e and A for OK and KI are also given in Table 2.1
at about the same volume fraction as AK. Note that KI is observed to be considerably more rigid
than the other two muds, as reflected by the e and A values for G1 and G2. Increasing stiffness
represented by G1 with increasing frequency in fact implies an approach towards a Voigt state
(very large G1). As shown elsewhere (Jiang, 1993), Eq. 2.4 and the Voigt model become
practically coincident for AK when the frequency exceeds about 10 Hz. Since this is an
excessively high value for the natural environment, the Voigt model in general represents an
oversimplified viscoelastic description of the prevailing physical state of mud.
2.4 CONCLUDING COMMENTS
We have presented Eq. 2.4 as an operational model describing mud rheology without a
formal proof of its validity or uniqueness of its form. Other models, each however necessarily
more complex than Eq. 2.4, may be sought since theoretically the coefficients a, 3 of Eq. 2.1
must be true constants if they are to represent a constitutive material property. Irrespective of
this limitation however, the data clearly emphasize the dependence of mud theological response
on frequency. This dependence is highlighted in Fig. 2.6, in which the magnitude of the
viscosity, /1e, is plotted against frequency for the three muds.
The yield behavior can be significant from the field perspective as the hydrodynamic
stresses can exceed Ty under some natural scenarios. When this occurs the rigidity drops
drastically and the response becomes more viscous, often leading to a liquefied state of mud
under continued wave action (Feng et al., 1992).
1in5
I 1 11111 I 1 f11111 I
KI
AK..
C 102
^ 101 rft
10,
S !\ I I I III I I I 1l iad I f I il
102 101 100 101
FREQUENCY, f (Hz)
Figure 2.6. Magnitude of viscosity pe versus frequency f for AK, OK and KI muds.
PART 3
APPLICATION OF A VISCOELASTIC WAVEMUD INTERACTION MODEL
SUMMARY: A wavemud interaction model that: 1) accounts for the finite height of water
waves, and 2) treats mud as a viscoelastic material characterized by two moduli of elasticity and
viscosity, has been used to simulate wave damping across the Mobile berm. Given measured
wave spectra offshore of the berm, the model simulates the inshore spectra arising from wave
passage over the berm. Mud rheology was determined through controlledstress rheometry. As
shown by the model and attested by the field data, the berm causes a significant amount of wave
damping, and effectively serves as a mitigator of potential wave impacts leeward of the berm.
Discrepancies between simulation and measurement arise at relatively high frequencies probably
due to the fact that a 2D model has been applied to a prototype situation in which the 3D
nature of the spectral wave field cannot be wholly ignored. In addition, a wider set of samples
than selected may be required to obtain a more representative, description of mud rheology at
this site where the dredged material is highly graded. It is shown that elsewhere, along the
southwestern coast of India, where the waves were seemingly aligned in the crossshore direction
and the sediment was comparatively uniform, a better agreement between measurement and
simulation can be achieved.
3.1 INTRODUCTION
Four noteworthy limitations of the wavemud interaction model presented in Part 1 are
contained in the assumptions: 1) shallow water wave condition, 2) small amplitude waves,
3) negligible elasticity of mud and 4) an inviscid water layer. Here a more advanced model in
which these assumptions have been removed is briefly presented. Details are found in Jiang
(1993). The introduction of a viscoelastic model for mud behavior is according to the description
presented in Part 2. The wavemud model is applied to the Mobile berm site described in Part
1 to simulate the observed attenuation of wave energy across the berm.
3.2 WAVEMUD INTERACTION
The 2D boundary value problem is defined as follows. Given the xdirection of wave
propagation and the vertical coordinate, z, relative to still water level (and positive upward), the
equation of flow continuity is
I + W 0 (3.1)
ax az
where uj(x,z,t) and wj(x,z,t) are the components of velocity, with j = 1 for water and 2 for
mud, and t is time. Starting with the equation of motion for a small disturbance in the
viscoelastic medium the x and the zcomponents of the equation of motion can be shown to be,
respectively (Kolsky, 1963; MacPherson, 1980):
Duj 1 aj a2uj + a2Uj (3.2)
Dt pj ax ax2 az
Dwj __ 1 Pj a2wj a2wj (3.3)
Dt pj xz e ax2 z 2
in which the symbol D denotes a total derivative, pj is the density, pj(x,z,t) is the pressure and
vej is the kinematic viscosity, which is a complex quantity for the mud layer. The pressure is
given by
t o (3.4)
pj = Pj + pjgz + pj
S 0, for j = 1 (35)
J (p2Pl)ghl, for j = 2
where hi is the mean water depth, p.t is the total pressure and g is the acceleration due to
gravity. The free surface is denoted by l(x,t) and the interface by 772(x,t), both with respect to
the corresponding still levels. In water the viscosity vel = v+el, where E1 is the turbulent eddy
viscosity, which in general depends on the wave Reynolds number, Rw = ab2/vl, defined in
terms of the amplitude of excursion, rb, of the fluid particle just above the boundary layer. The
value of ve2 depends on the theological model described in Part 2.
Ten boundary conditions are required to solve Eqs. 3.1, 3.2 and 3.3 for the water surface
profile and the velocity field. At the fixed bottom of mud layer, i.e. at z = (hl+h2), where h2
is the mean mud depth, the horizontal (u2) and the vertical (w2) velocities are nil. At the
interface, z = h1+r12, equality of velocities, normal stress and shear stress as well as the
interfacial kinematic condition respectively require that u1 = u2, w1 = w2, and
t awl t aw2 (3.6)
Pl 2plvl  = P2 2P2e2 z
aul aw] u2 u w2 (3.7)
pe, l + 2 +
ei e z ax I az ax
2= w W2 (3.8)
at= W
At the free surface, z = rt, the dynamic and the kinematic conditions are
p 2ple 0 (3.9)
Iel au1 awl (3.10)
Swl, (3.11)
at W
The free surface is assumed to be harmonically forced at x = 0, neglecting any influence of
atmospheric pressure variation (Eq. 3.9) or wind stress (Eq. 3.10).
The solution of the boundary value problem is sought through the perturbation method
by expressing each variable as a convergent power series of a small perturbation parameter, such
as the water surface slope. Thus for example, rj = r])+ 2)+ .... etc., wherein the
perturbation parameter is absorbed into the function. Substituting this and the corresponding
expressions for u., wj, pj and p.t into Eqs. 3.1, 3.2 and 3.3 and the boundary conditions, the first
order (linear) and the second order problems are specified. The first order problem solution is
straightforward, and has been stated by Dalrymple and Liu (1978), given a harmonic
representation of all the variables, e.g. i.(1) = aj('exp[i(kxat)] etc. Here a(1) is the wave
amplitude, superscript (1) denotes a first order quantity, i = (1)1/2 and k = kr+iki is the
complex wave number. The quantity kr is the wave number and ki = ln(ax/ao)/x is the wave
attenuation coefficient, where ao is the wave amplitude at x = 0 and ax is the amplitude at any
distance x. Note that the advective terms, being of second order, are not included in the first
order case. Thus, the total derivatives in Eqs. 3.2 and 3.3 are replaced by the corresponding
partial ones. Also, by virtue of the first order approximation the interface and the free surface
must be specified as z = h1 and z = 0, respectively.
Corresponding to the first order variables the second order ones are assumed to be of the
form (2) = a.(2)exp[i2(kxat)] etc., in which aj(2) is the amplitude of j(2), (2) denotes a second
order quantity and k is obtained from the first order solution. These variables are determined
in terms of the corresponding first order solutions. Analytic solutions for the water surface
profile and waveinduced velocities in water and in mud as functions of the viscosity, vej, and
other parameters are found in Jiang (1993).
Finally, the Lagrangian residual mass transport velocity, uL(Z), or the Stokes' drift is
obtained from (Sakakiyama and Bijker, 1989; Jiang, 1993):
L = u2 + 2dt + d w2dt (3.12)
Tx J 0 2 az Jo
where u2 is the Eulerian mass transport velocity. The significance of the Lagrangian mass
transport velocity has been discussed qualitatively in Section 1.2.
3.3 SOME MODEL APPLICATIONS
The model was tested against data obtained in a laboratory flume using AK mud
described in Part 2. The relationship between the dynamic viscosity, /e and the forcing
frequency, f, for this mud is given in Fig. 2.6. The flume tests occurred at Reynolds number,
Rw, less than 7,500, so that turbulence in the water column was ignored for purposes of
simulation (eq = 0). An illustrative result is shown in Fig. 3. la in which the modelcalculated
wave attenuation coefficient, ki, has been plotted against frequency a (=27f), and compared
with flume data. Values of ki and the trends of its variation with frequency are reasonably well
predicted. The physical significance of the ki maxima is noted in Section 1.3. In Fig. 3.1b the
vertical profile of the amplitude of the horizontal acceleration, 12, is shown. The data were
obtained within the mud layer using a miniature accelerometer. The model is able to reproduce
the profile to a reasonably good degree. Notice that the predicted profile exhibits a drastic
reduction in the acceleration in the vicinity of the mudwater interface, due to the high degree
of energy dissipation within the mud.
As noted in Part 1, nearshore mud banks along the southwest coast of Kerala in India are
highly mobile in the sense that they arrive close to the shoreline within days of the onset of the
southwesterly monsoon (in May or June), remain there as long as the monsoon waves continue,
and slide into deeper offshore waters as the monsoon wanes (in August). Mathew (1992)
measured wave spectra at two sites shown in Fig. 1.5. During the fair weather measurement
(e.g. Fig. 1.6a) the mud bank was seaward of the offshore site and the bottom over entire stretch
between the two sites was sandy. During monsoon (Fig. 1.6b) about 1 m thick mud covered
most of the stretch between the two stations, although waves at the offshore site were not
significantly influenced by the mud. The relationship between the magnitude of the viscosity,
ILe, and wave frequency for this mud (KI) at the natural solids volume fraction of 0.12 is given
in Fig. 2.6.
The model was applied on a frequencybyfrequency basis by selecting the eddy
diffusivity, e1 = 0.0075 m2/s, a typical oceanic value, and taking the offshore wave spectrum
as input at x = 0. The fair weather case (Fig. 1.6a) was treated by selecting arbitrarily high
values of G1 and G2, in order to simulate a rigid bottom. Results shown in Figs. 3.2a,b indicate
that the inshore wave spectra are simulated reasonably well for the fair weather as well as
monsoon (Fig. 1.6b) scenarios. 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
F
z
LL
UJ
I
z
w
UJ
I
0.4 
0.2 
0.0 L
0
ANGULAR FREQUENCY, a (rad/s)
Figure 3. la. Wave attenuation coefficient against frequency for two AK mud depths, 12 cm and
18 cm. The water depth is 16 cm.
0.0
S0.2
U.
S0.4
w
, 0.6
S0.8
0.8
0.05 0.10 0.15 0.20
0.25
ACCELERATION AMPLITUDE (m/s2)
Figure 3. lb. Comparison between measured and modelcalculated profiles of the amplitude of
horizontal acceleration in a watermud (AK) system in a flume. The water depth was 16 cm,
mud thickness 18 cm, water surface forcing wave amplitude 0.5 cm, wave frequency 1 Hz and
4 = 0.12.
AK Mud
h2 = 18 cm
h2 = 12 cm
I I I I
0.00
04* (aI I I
E 0.4 (a) Alleppey, India
>Measurement
I_
C 0.3
z
L \ Simulation
0
> 0.2
u)
S0.0 I I
S 0.0 0.1 0.2 0.3 0.4
FREQUENCY (Hz)
E 0.4 (b) Alleppey, India
!
Cn 0.3
z
> 0.2
z 0.1
Lu
> 0.0 I I
S 0.0 0.1 0.2 0.3 0.4
FREQUENCY (Hz)
Figure 3.2. Comparison between measured and modelsimulated inshore wave spectra off
Alleppey in Kerala, India: (a) fair weather condition (mudbank absent), and (b) monsoon
condition (mudbank present).
onset of monsoon, even though no actual mud velocity measurements are available for
comparison purposes (Mathew, 1992).
3.4 MOBILE BERM MUD RHEOLOGY
Three representative mud (MB) samples were tested in the CarriMed CSL rheometer in
accordance with the procedures given in Jiang (1993), and summarized in Part 2. These samples
had the same sediment composition (and with a granular density of 2,650 kg/m3) but differed
in density; the solids volume fractions (4) being 0.07. 0.11 and 0.17. These values of 4, cover
the typical range of the mud density under field conditions.
The dependence of the shear moduli, G1 and G2, and the viscosity, p, on the forcing
frequency, f, is shown in Fig. 3.3 for 4 = 0.17. The applied stress amplitude, rT, was varied
from 4 to 25 Pa, a fairly wide range. As in the case of Fig. 2.5 for AK mud, the dependence
of the viscoelastic parameters on stress can be considered to be of secondary importance in
comparison with their dependence on f. Similar results were obtained for 4 = 0.07 and 0.11.
Thus, in accordance with Eq. 2.6 the corresponding coefficients, e and A, are given in
Table 3.1.
Table 3.1. Coefficients of Eq. 2.6 for MB mud
G1(Pa) G2(Pa) p(Pa s)
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
In order to examine the dependence of the viscoelastic parameters on mud density, values
of e and A for G1, G2 and 1 are plotted against 4 in Figs. 3.4a,b. It is observed that the
parameters increase monotonically with 4q, although the data are insufficient (in terms of the
range of 4 covered) for drawing further inferences concerning the observed trends.
3.5 WAVE ATTENUATION OVER MOBILE BERM
The wavemud interaction model was used to simulate the inshore wave spectra at the
Mobile berm site, given measured offshore spectra in Figs. 1.8a,b. The computations were done
on a frequencybyfrequency basis. At this site the actual berm dimensions differ from those
given in Section 1.1 that are meant to indicate design conditions. Yet, the 6 m berm height (=h2
= hM) and the 6.5 m water depth (= h1 = h) are believed to be reasonable representative
105 r i i i Ii i i
_
104
6 0
To (Pa)
103 
3 MB Mud A 4.0
0 : = 0.17 0 25.0
1042 i iiiiii i iiiiii i i i gii
S10
103
104
a 00
0 0
100
CI) 102 o 0
O2
I I I I I I III I I I I 1 il i I I I I I IIT
> 02 101 10 0 101
I0
S105
102 101 100 101
FREQUENCY, f (Hz)
Figure 3.3. Experimentally determined variation of the viscoelastic parameters, G1, G2 and /,
with the forcing frequency, f, for MB mud.
0.2
(a) (b)
0 00 i
0.05 0.10 0.15 ()
G,
0.2
A
0 0 0.4
2 0 0.6
0 o 0.8
0.05 0.10 0.15 4)
0
2 1.0
Figure 3.4. Experimentally determined variation of the coefficients in Eq. 2.6 with the solids
volume fraction, 4, for MB mud: (a) E versus 4b, and (b) A versus 4.
values. However, bathymetric surveys (Dredging Research Technical Notes, 1992) suggest that
there are significant irregularities in the berm crest configuration in plan, and therefore it is
difficult to determine a characteristic berm width. Furthermore, the effective berm width for
calculating the degree of wave attenuation depends on the direction of wave approach. For the
present purpose a width of 140 m was selected as a representative value, giving some allowance
for the berm slopes. The turbulent eddy diffusivity, E1, to which the model results are not
sensitive, was selected to be 0.0075 m2/s, as in the case of the Alleppey application.
Comparisons between simulated and measured inshore spectra are given in Figs. 3.5a,b
for solids volume fraction, 04 = 0.11 selected as the representative mud density. 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 greater than 4 s. In that connection 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 is the case
here as well. Compare, for example, the measured offshore spectrum with the simulated inshore
one in Fig. 3.5a. 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
C4 0.20
/Offshore (Measured)
v
U)
Z 0.15
Inshore (Measured)
UJ 0.10 I Inshore (Simulated)
z
1 o.o0 ,, 
1.0
Mobile Berm, Alabama (b)
8/7/88
T 0.8
E Offshore (Measured)
I
S0.6
z
W
Inshore (Simulated)
Cc 0.4
UJ
z
W
S0.2 Inshore (Measured)
0.0
0.0 0.1 0.2 0.3 0.4 0.5
FREQUENCY (Hz)
Figure 3.5. Comparison between measured and modelsimulated inshore wave spectra at the
Mobile berm site corresponding to two offshore wave conditions: (a) Hma = 0.9 m (see
Fig. 1.8a), and (b) Hmax = 1.5 m (see Fig. 1.8b). Measured offshore spectra have been
included for reference.
40
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 that context, with reference to Fig. 1.7 showing the locations of the wave stations, we
recognize that it is uncertain if during the times of wave data collection the predominant
direction of wave propagation was along the direction of the line connecting the two
measurement stations. Thus, in general it can be surmised that the waves that arrived at the
inshore station were not in full alignment with those at the offshore location. This "deviation"
is one probable cause of the discrepancy between the measured and simulated spectra for
frequencies greater than 0.25 Hz.
The second probable cause of the discrepancy arises from the variability in the sediment
composition at the berm. This variability reflects the heterogeneity in the dredged material. In
order to determine if the choice of the solids volume fraction (0.11) was significant, we ran the
model with 4 = 0.07 and 0.17. It was found that the spectral shape was weakly dependent on
) over this range. Thus the discrepancy between the measured and the simulated spectra cannot
be explained on that basis. On the other hand, there may have been a measurable effect of the
grain size (hence mineral composition) on the rheology. The MB mud sample was actually a
composite of four samples obtained from different locations on the berm, with a median grain
size of about 8 pm for the composite sample. This size is roughly twice the prevailing median
value of 15 im over the berm. Note that in general cohesion, and hence the dissipative property
of mud, increases rapidly with decreasing grain size as the grain size decreases below about 20
ttm. Thus, the MB mud sample was most probably more dissipative than a more representative
mud sample would have been. This difference may have contributed to the higher than measured
wave energy reduction in the simulation in the high frequency range of the wave spectrum.
3.6 CONCLUDING COMMENTS
The wavemud interaction model generally predicts the degree of wave damping at the
Mobile berm with a reasonable degree of accuracy with the exception of that portion of the
spectrum corresponding to frequencies greater than 0.25 Hz. One probable cause of the
discrepancy at high frequencies is the 3D nature of the wave field. In other words, it is likely
that the measured inshore wave spectrum was to some extent determined by waves that did not
arrive there from the direction of the offshore wave station. Another likely cause of the
discrepancy is the rheology of the mud sample, which was more dissipative than what a more
representative sample for the entire berm would have been. Elsewhere, e.g. off Alleppey, where
such a 3D condition may not have contaminated the data and the sediment was comparatively
uniform, model simulation better agreed with the measurements.
The influence of changing mud elasticity on wave attenuation is noteworthy. Thus, for
instance, if the mud were assumed to be a Voigt material by selecting G1 to be a very large
value, the spectral shape would be affected (Jiang, 1993). Also note that the theological
characterization considered here did not account for thixotropic yield, which could also influence
the simulation measurably.
The effect of changing rheology can be briefly examined as follows. In Fig. 3.6 it is
observed that changing the MB mud into a Voigt material would cause greater energy dissipation
than otherwise. This would be so as a result of the increase in viscous dissipation relative to the
Voigt elasticity (G2). Note that in this example the effect of changing the rheology is not
dramatic, because the theological model is not too different from Voigt, as a result of the high
value of G1 (Fig. 3.3). Note further that if the bottom were assumed to be rigid (very high
values of G1 and G2), practically no damping would occur over the width of the berm as
indicated by McLellan et al. (1990) by using a rigid bottom wave propagation model. The
present model yielded the same result, as in the case of model application to the Alleppey
mudbank shown in Fig. 3.2a.
It needs to be emphasized that the Mobile berm appears to be effective in damping
approaching water waves significantly due to the highly dissipative nature of the mud bottom.
In that context, the model is a useful tool for examining the essential physics underlying the
dissipative mechanism, which is almost entirely embodied in the theological constitutive
relationship for the mud.
In Part 1 we presented an application of the shallow water model to calculate design berm
crest elevation and water depth. That model relies on less input information than the model
presented here, and is amenable to a graphical presentation of berm design calculations. With
reference to the present model a similar graphical solution is not practicable. Yet, because it
" 0.20
I
i)
Z 0.15
LU
>
W 0.10
Z
z
LU
0.05
FREQUENCY (Hz)
Figure 3.6. Comparison between measured and modelsimulated inshore wave spectra at the
Mobile berm site. The mud rheology has been changed to a Voigt material description.
better incorporates the physics of the wavemud interaction problem, it can be potentially used
for berm design calculations, under conditions for which the assumptions underlying the shallow
water model are not valid. The model can also be used to assess potential wave impactswhere
a preexisting berm or a bank is removed naturally or otherwise, thus exposing a hard bottom
over which wave dissipation would be reduced.
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