• TABLE OF CONTENTS
HIDE
 Front Cover
 Report documentation page
 Title Page
 Acknowledgement
 Table of Contents
 Synopsis
 Introduction and physical...
 Field site
 Experiments
 Water and mud motions
 Response of linearized fluid mud-water...
 Low frequency signature
 Concluding remarks
 References
 Appendix A: Influence of water...
 Appendix B: Operation of the...
 Appendix C: Measurement of mud...
 Appendix D: Inviscid-viscid flow...
 Appendix E: Measured spectra in...






Group Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; no. 90/008
Title: Some field observations on bottom mud motion due to waves
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 Material Information
Title: Some field observations on bottom mud motion due to waves
Series Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; no. 90/008
Physical Description: Book
Creator: Mehta, Ashish J.
Publisher: Coastal and Oceanographic Engineering Department, University of Florida
Publication Date: 1990
 Subjects
Subject: Coastal Engineering
Mud
Water waves
 Notes
Funding: This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
 Record Information
Bibliographic ID: UF00080977
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.

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Table of Contents
    Front Cover
        Front Cover
    Report documentation page
        Unnumbered ( 2 )
        Unnumbered ( 3 )
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
        Page v
        Page vi
        Page vii
    Synopsis
        Page viii
        Page ix
        Page 7
    Introduction and physical perspective
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
    Field site
        Page 8
        Page 7
        Page 9
    Experiments
        Page 10
        Page 9
        Page 11
        Page 12
    Water and mud motions
        Page 13
        Page 14
        Page 15
    Response of linearized fluid mud-water system
        Page 16
        Page 15
        Page 17
        Page 18
        Page 19
        Page 20
    Low frequency signature
        Page 21
        Page 22
        Page 23
        Page 20
    Concluding remarks
        Page 24
        Page 23
    References
        Page 25
        Page 26
        Page 27
    Appendix A: Influence of water level on mud area subject to resuspension
        Page 28
        Page 29
    Appendix B: Operation of the accelerometer
        Page 30
        Page 31
    Appendix C: Measurement of mud viscosity
        Page 32
        Page 33
    Appendix D: Inviscid-viscid flow problem solution
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
    Appendix E: Measured spectra in Tests 1 and 2
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
Full Text




UFL/COEL-90/008


SOME FIELD OBSERVATIONS ON BOTTOM MUD
MOTION DUE TO WAVES





by



Ashish J. Mehta
Feng Jiang


Sponsors:

South Florida Water Management District (SFWMD)
West Palm Beach, Florida
U.S. Army Engineer
Waterways Experiment Station (WES)
Vicksburg, Mississippi


October, 1990






REPORT DOCUMENTATION PAGE
1. Report No. 2. 3. Recipient's Accesoioo go.
UFL/COEL-90/008

4. Tit1l and Subtitle 5. Report Data
SOME FIELD OBSERVATIONS ON BOTTOM MUD MOTION October, 1990
DUE TO WAVES 6.

7. Author(@) 8. Performia Organization Report o.
Ashish J. Mehta UFL/COEL-90/008
Feng Jiang U..
9. Performing OrganizatiLoo am and Address 10. Project/Task/Mork Uit So.
Coastal and Oceanographic Engineering Department Task 4.4b*
University of Florida 11. contract or crant no.
336 Weil Hall DACW39-89-M-4639**
Gainesville, FL 32611 13. Typ of Iport
12. Sponsoring Organization Name ad Address
South Florida Water Management District (SFWMD) Final
West Palm Beach, Florida
U.S. Army Engineer Waterways Experiment Station (WES)
Vicksburg, Mississippi 14.
15. Supplementary Notes
*Lake Okeechobee Phosphorus Dynamics Study (SFWMD)
**Monitoring Fluid Mud Generation (WES)

16. Abstract (Synopsis)


The behavior of soft mud under progressive, non-breaking wave action has been
briefly examined in the vicinity of the Okeechobee Waterway, Florida. The main
objective was to demonstrate in the field that under wave conditions that are too
mild to cause.significant particle-by-particle resuspension, soft mud layers on the
order of 20 cm thickness can undergo measurable oscillations induced by wave
loading. Among other matters, such a motion may have implications for the rates of
diffusive exchange of nutrients and contaminants between the bottom and the water
column, and the formation and upward transport of gas bubbles, which are ubiquious
in the mud in the study area. Continued mud motion can also retain the mud in a
fluidized state, thereby enhancing its availability for resuspension during episodic
events.

The chosen field site was in the shallow littoral margin of Lake Okeechobee,
where the water depth was on the order of 1.5 m over a 0.5 m thick muddy substrate.
During two experiments the water waves, induced orbital velocities in the water
column and corresponding accelerations within the bottom mud layer were measured. In
addition, bottom density profiles were obtained. A simple, two-layered wave

Continued -


17. Originator's Key Words 18. Availbility Statment
Fine sediment Sedimentation
Mud motion Surf beat
Okeechobee Waterway Water quality Available
Resuspension Waves

19. U. S. Security Classif. of the Report 20. U. S. Security Clessif. of This Page 21. No. of Pages 22. Price
Unclassified Unclassified 76










propagation model which considers the water column to be inviscid and the mud layer
to be a high viscosity fluid has been used to aid in data interpretation. Prior
evidence indicates that 5-20 cm thick bottom surficial mud layer, which is rich in
organic content (40% by weight), persists in the fluidized state over much of the
area of the lake consisting of muddy bottom. In the first test in which wind wave
frequency was on the order of 0.4 Hz and significant wave height around 10 cm, wave
coherent mud motion was measured 20 cm below the mud-water interface, where the mud
density was 1.18 gm/cm3. In the second test similar motion occurred 5 cm below the
interface.

Given the wave energy spectrum, the wave model approximately simulates both the
water velocity spectrum as well as the mud acceleration spectrum, and highlights the
fact that wave attenuation is strongly frequency dependent. Deviations between
prediction and measurement are pronounced in the high frequency range of mud
accelerations wherein the shallow water assumption inherent in the model breaks
down. The muddy bottom causes waves to attenuate much more significantly than what
would occur over a hard bottom. Model results indicate wave damping coefficients on
the order of 0.005 m-1 in the shallow areas. These high values (compared with -10-5
m over rigid beds) may explain why the waves arriving at the test site were only a
quarter as high as those that might be expected if the lake bottom were wholly
rigid.

A low frequency, long wave signature (e.g. at about 0.04 Hz in the first test),
was characteristic of measured spectra. This signal was enhanced in the mud relative
to the forcing signal (at 0.4 Hz) due to the dependence of wave attenuation on
frequency, and led to horizontal mud displacements (twice the amplitude) on the
order of 2 mm at 20 cm depth in the first test and 5 cm in the second. Since the
dominant seiching frequency in the lake is around 10- Hz, a different cause must be
found for the occurrence of the long wave. Although an unambiguous causative
mechanism is not entirely apparent, it is suggested that the long wave signal is
akin to surf beat characteristic of water level fluctuations at open coasts. The
compliant bottom allows for the signal to be transmitted into the muddy substrate.
This wave causes the mud to oscillate very slowly, thereby contributing to its
mobility.









UFL/COEL-90/008


SOME FIELD OBSERVATIONS ON BOTTOM MUD MOTION DUE TO WAVES





Ashish J. Mehta
Feng Jiang





Coastal and Oceanographic Engineering Department
University of Florida
Gainesville, FL 32611


October, 1990









ACKNOWLEDGEMENT


This work was supported by the South Florida Water
Management District (SFWMD), West Palm Beach, as a part of the
Lake Okeechobee Phosphorus Dynamics Study (Task 4.4b), and by the
U.S. Army Engineer Waterways Experiment Station (WES), Vicksburg,
MS (Contract DACW39-89-M-4639 titled Monitoring Fluid Mud
Generation). Thanks are due to Brad Jones and Dave Soballe of
SFWMD and Allen Teeter of WES for project management assistance.
Participation by Sidney Schofield, Nana Parchure and Kyu-Nam
Hwang in different phases of the study is acknowledged.











TABLE OF CONTENTS


ACKNOWLEDGEMENT . . . . .

LIST OF TABLES . . . . .

LIST OF FIGURES . . . . .

SYNOPSIS . . . . . .

I. INTRODUCTION . . . . .

II. A PHYSICAL PERSPECTIVE . . .

III. FIELD SITE . . . . .

IV. EXPERIMENTS . . . . .

V. BOTTOM MUD CHARACTERISTICS . . .

VI. WATER AND MUD MOTIONS . . . .

VII. RESPONSE OF LINEARIZED FLUID MUD-WATER SYSTEM .

VIII. LOW FREQUENCY SIGNATURE . . .

IX. CONCLUDING REMARKS . . . .

X. REFERENCES . . . . .


. ii

. iv


. .viii
.... viii



.... l1






. 13

. 15

... .....20

* 23

* 25


APPENDICES


A INFLUENCE OF WATER LEVEL ON MUD AREA SUBJECT TO

RESUSPENSION . . . .

B OPERATION OF THE ACCELEROMETER . .

C MEASUREMENT OF MUD VISCOSITY . . .

D INVISCID-VISCID FLOW PROBLEM SOLUTION . .

E MEASURED SPECTRA IN TESTS 1 AND 2 . .


iii


. .


. .


. .









LIST OF TABLES


TABLE
1 Test parameters (depths and elevations). . ... 11
A.1 Variation of erodible mud area with relative water
level. . . . . ... . .29









LIST OF FIGURES


FIGURE
1. Schematic of mud bottom response to waves in terms of
vertical sediment density and velocity profiles (after
Mehta, 1989) . . . . . 42
2. Two-layered water-fluid mud system subject to
progressive wave action . . . .. 42
3a. Bathymetric map of Lake Okeechobee. Depths are
relative to a datum which is 3.81 m above msl (NGVD) 43
3b. Mud thickness contour map of Lake Okeechobee (after
Kirby et al., 1989) . . . . 44
4. Lake area with mud bottom subject to wave action as a
function of water level relative to datum. . .. .45
5. Tower used in field tests: a) elevation view, b) plan
view . . . . . . 46
6. A view of the field tower. . . . ... 47
7. Tower and instrumentation assembly being deployed
at the site. . . . . . 47
8. Measurement system in place together with the data
acquisition system . . . . .. 48
9. Bottom core from test 1 is frozen in a mixture of
dry ice and alcohol and cut into 6-8 cm long
pieces. Note the clearly defined mud-water interface 48
10. Relationship between dynamic viscosity and density
for Okeechobee mud . . . . .. 49
lla. Mud density profile at the site during test 1. ... .50
lib. Mud density profiles at the site during test 2 ... 50
12a. Variation of significant wave height during test 1 51
12b. Variation of modal wave frequency during test 1. ... .51
13a. Wave energy spectrum at 1 hr, test 1 . ... .52
13b. Water velocity spectrum at 1 hr, test 1. . ... 52
14. Variation of relative direction of water velocity
during test 1. . . . . 53
15. Time-variation of water velocity amplitude variance
during test 1. . . . . ... .. .53









16a. Time-variations of the variances of horizontal and
vertical mud accelerations during test 1 . .. 54
16b. Variations of modal frequencies of horizontal and
vertical mud accelerations during test 1 . 54
16c. Horizontal mud acceleration spectrum at 1 hr, test 1 55
17a. Model calculated and measured water velocity spectra
at 1 hr, test 1. . . . .... .55
17b. Model calculated and measured mud acceleration spectra
at 1 hr, test 1 . . . . 56
18a. Wave energy density spectrum at 5 hr, test 1 . .. .57
18b. Model calculated and measured water velocity spectra at
5 hr, test 1 .. . . . . 57
18c. Model calculated and measured horizontal mud
acceleration spectra at 5 hr, test 1 . . .. 58
19. Dominant long wave frequency variation during test 1 59
20. Wave energy spectrum showing short period forcing at
two frequencies and forced long wave . ... .60
21. Short period forcing wave and forced long wave derived
from water level measurement at 1 hr, test 1 . .. .61
B.1 A view of the plexiglass "boat" together with the
accelerometer (not visible). The boat length is 26 cm.
Pen is for length reference only . . ... .62
B.2 Scale measured versus calculated (from acceleration)
wave orbital displacements amplitudess) based on
dynamic testing of accelerometer . . .. 62
C.1 Relationship between applied stress and rate of
shearing for Okeechobee mud; data for mud density
of 1.005 g/cm3 . . . .. . 63
C.2 Relationship between applied stress and rate of
shearing for Okeechobee mud; data for mud density
of 1.02 g/cm3. . . . . . 63
C.3 Relationship between applied stress and rate of
shearing for Okeechobee mud; data for mud density
of 1.04 g/cm3. . . . . 64
C.4 Relationship between applied stress and rate of
shearing for Okeechobee mud; data for mud density
of 1.08 g/cm3. . . . . .. 64


1









C.5 Relationship between applied stress and rate of
shearing for Okeechobee mud; data for mud
density of 1.1 g/cm . . . .
C.6 Relationship between mud viscosity (relative to
water) and density at "high" and "low" rates
of shearing . . . . .
D.1 Dispersion relationship based on the inviscid-
viscid model . . . . .
D.2 Wave attenuation relationship based on the inviscid-
viscid model . . . . .
D.3 Simulated profiles of velocity amplitude, u, for
different values of X using parameters from test 1
D.4 Simulated profiles of the phase of u, corresponding
to Fig. D.3 . . . . .
E.1 Wave energy spectra, test 1, 0-3 hrs . .
E.2 Wave energy spectra, test 1, 4-7 hrs . .
E.3 Water velocity spectra, test 1, 0-3 hrs. . .
E.4 Water velocity spectra, test 1, 4-7 hrs. . .
E.5 Mud acceleration spectra, test 1, 0-3 hrs. . .
E.6 Mud acceleration spectra, test 1, 4-7 hrs. . .
E.7 Wave energy spectrum at 1800 hr, test 2. . .
E.8 Horizontal velocity spectrum at 1800 hr, test 2. .
E.9 Vertical velocity spectrum at 1800 hr, test 2. .
E.10 Horizontal acceleration spectrum at 1800 hr, test 2.
E.11 Vertical acceleration spectrum at 1800 hr, test 2. .


vii


. 65


. 67


. 67
S 68
. 69
S. 70
. 71
. 72
S. 73
. 74
. 74
S. 75
. 75
. 76


. .


I I









SYNOPSIS


The behavior of soft mud under progressive, non-breaking
wave action has been briefly examined in the vicinity of the
Okeechobee Waterway, Florida. The main objective was to
demonstrate in the field that under wave conditions that are too
mild to cause significant particle-by-particle resuspension, soft
mud layers on the order of 20 cm thickness can undergo measurable
oscillations induced by wave loading. Among other matters, such a
motion may have implications for the rates of diffusive exchange
of nutrients and contaminants between the bottom and the water
column, and the formation and upward transport of gas bubbles,
which are ubiquious in the mud in the study area. Continued mud
motion can also retain the mud in a fluidized state, thereby
enhancing its availability for resuspension during episodic
events.
The chosen field site was in the shallow littoral margin of
Lake Okeechobee, where the water depth was on the order of 1.5 m
over a 0.5 m thick muddy substrate. During two experiments the
water waves, induced orbital velocities in the water column and
corresponding accelerations within the bottom mud layer were
measured. In addition, bottom density profiles were obtained. A
simple, two-layered wave propagation model which considers the
water column to be inviscid and the mud layer to be a high
viscosity fluid has been used to aid in data interpretation.
Prior evidence indicates that 5-20 cm thick bottom surficial mud
layer, which is rich in organic content (40% by weight), persists
in the fluidized state over much of the area of the lake
consisting of muddy bottom. In the first test in which wind wave
frequency was on the order of 0.4 Hz and significant wave height
around 10 cm, wave coherent mud motion was measured 20 cm below
the mud-water interface, where the mud density was 1.18 gm/cm3.
In the second test similar motion occurred 5 cm below the
interface.
Given the wave energy spectrum, the wave model approximately
simulates both the water velocity spectrum as well as the mud


viii









acceleration spectrum, and highlights the fact that wave
attenuation is strongly frequency dependent. Deviations between
prediction and measurement are pronounced in the high frequency
range of mud accelerations wherein the shallow water assumption
inherent in the model breaks down. The muddy bottom causes waves
to attenuate much more significantly than what would occur over a
hard bottom. Model results indicate wave damping coefficients on
the order of 0.005 m1 in the shallow areas. These high values
(compared with ~10- m- over rigid beds) may explain why the
waves arriving at the test site were only a quarter as high as
those that might be expected if the lake bottom were wholly
rigid.
A low frequency, long wave signature (e.g. at about 0.04 Hz
in the first test), was characteristic of measured spectra. This
signal was enhanced in the mud relative to the forcing signal (at
0.4 Hz) due to the dependence of wave attenuation on frequency,
and led to horizontal mud displacements (twice the amplitude) on
the order of 2 mm at 20 cm depth in the first test and 5 cm in
the second. Since the dominant seiching frequency in the lake is
around 10-4 Hz, a different cause must be found for the
occurrence of the long wave. Although an unambiguous causative
mechanism is not entirely apparent, it is suggested that the long
wave signal is akin to surf beat characteristic of water level
fluctuations at open coasts. The compliant bottom allows for the
signal to be transmitted into the muddy substrate. This wave
causes the mud to oscillate very slowly, thereby contributing to
its mobility.









The overall objective of the field investigation was to
record oscillatory motion of fluidized mud in response to wind-
generated waves at a shallow site with a muddy substrate. The
field site and the experiments are noted below.


III. FIELD SITE
The main requirements for the field site were: 1) wave-
dominated environment, 2) shallow water, and 3) a soft muddy
bottom with sizeable thickness of fluidized mud. These conditions
are approximately met in the southeastern part of Lake Okeechobee
close to the shoreline (-1 km offshore) in the vicinity of the
Okeechobee Waterway, a part of the Intracoastal Waterway system.
Fig. 3a shows depths in this rather shallow lake, and
Fig. 3b shows the bottom mud thickness. Depths in Fig. 3a may be
considered to be relative to the top of the mud. The depth datum
is 3.81 m above NGVD, but the actual water level in the lake is
subject to significant variation imposed by the inflows and
outflows which are controlled. Thus, for example, during a field
excursion to collect bottom sediment samples in March, 1988
(Hwang, 1989) the actual water level was about 1.2 m higher than
the datum.
The variation of water level implies the likelihood of a
corresponding variation in the mud bottom area which is
influenced by wave action. Assuming typical storm-induced wave
characteristics and their erodibility potential as represented by
the critical shear stress for erosion, a water depth of 3.4 m can
be calculated (see Appendix A) as the critical depth such that
the bottom will erode only if the actual depth is equal to or
less than this critical depth. Based on this criterion, Fig. 4
shows the lake area influenced by waves at different water levels
(arbitrarily selected to be -1.0 m to +1.5 m relative to datum).
It is observed that when the lake level is less than 0.5 m below
datum, the entire mud bottom area of 528 sq. km is subject to
wave action. On the other hand, when the level is, for example,
1.5 m above the datum, the affected area is reduced to 48 sq. km.
The shape of the curve further implies that the bottom area
influenced by wave action is most sensitive to water level in the









SOME FIELD OBSERVATIONS ON BOTTOM MUD MOTION DUE TO WAVES


I. INTRODUCTION
It is generally well recognized that in shallow, episodic
coastal or lacustrine environments with muddy beds, reworking of
mud by waves causes the bottom to become loose, with looseness
persisting as long as waves continue and thereafter, until the
bottom material dewaters sufficiently to lead to hardening under
calm conditions. Laboratory evidence shows that waves cause the
mud bed to fluidize under cyclic loading, which breaks up the
structural matrix of the bed held together by cohesive, inter-
particle bonds. Furthermore, fluidization may occur without much
entrainment of sediment in the water column, in which case no
significant change in the bottom mud density occurs either (Ross
and Mehta, 1990). In the limiting case of no resuspension (i.e.
particle-by-particle erosion of the mud interface and upward
entrainment of the eroded particulate matter), and therefore no
density change of the bottom material, measurement of sediment
concentration at different elevations would yield no evidence of
the change of state of the mud from a cohesive bed to a fluid-
supported slurry. Yet this change of state has obvious
implications in bottom boundary layer related phenomena,
including for example: 1) the availability of fluidized mud for
potential resuspension by current or strong wave action, and 2)
possible change in the effective permeability or resistance to
diffusion, leading to corresponding changes in the exchange of
nutrients or contaminants between the bottom and the water
column. It is therefore relevant to examine the issue of mud
motion by waves in terms of the nature of motion that results
from wave action, and mud properties that influence the results.
In this study the problem was examined from the following
physical perspective.


II. A PHYSICAL PERSPECTIVE
A simple physical perspective is chosen to deal with a
rather complex problem which is characterized by time-dependent
changes in mud properties with continued wave action. Although









such changes have been tracked to some extent in laboratory
experiments, field evidence is scarce due to evident problems in
deploying requisite transducers. Furthermore, the basis for any
theoretical examination of the time-variability of such
properties as mud shear strength and rheology is presently
inadequate. In treating the problem these limitations impose
certain operational limitations in data gathering and analytic
constraints in data analysis, which must be borne in mind as in
the case of the following development.
In the way of a general description of the problem, it is
instructive to consider Fig. 1, in which sediment density (p)
profile and the horizontal component of the wave-induced velocity
amplitude (u,) in the water column and bottom mud are depicted in
a somewhat idealized manner. With regard to the density profile,
the important feature to recognize is the characteristic
horizontal layering of the system. In the upper water column, in
which pressure and inertia forces are dominant in governing water
motion and the flow may be treated as essentially irrotational
(ignoring the relatively thin wave boundary layer ref.), the
sediment concentration tends to be typically low, say on the
order of 0.1 g/l or less. Thus the suspension density is close to
that of water. The lower boundary of the layer is characterized
by a rather significant gradient in concentration, or lutocline,
below which the concentrations of the fluidized mud are
considerably higher, on the order of 10 to 200 g/l (density
range: 1.01 to 1.12 g/cm3 in fresh water).
Below fluidized mud is the cohesive bed having yet higher
concentrations. Laboratory observations by Maa (1986) and Ross
(1988), and theoretical work by Foda (1989) show that the wave
orbits can penetrate the bed, thereby leading to elastic
deformations of the bed. Under continued wave loading such
deformations, coupled with a buildup of excess pore pressure, can
cause fluidization, and this is in fact one way by which the
thickness of the fluidized layer increases, starting, say, from a
two-layered system of a porous solid bed and a clear water column
at incipient wave motion (Ross and Mehta, 1990).









Recognizing that, due to the generally low rates of upward
mass diffusion above the wave boundary layer, and therefore low
observed concentrations of suspended sediment over most of the
water column, the problem of mud motion by waves can be
conveniently considered to be practically uncomplicated by the
effects of particle-by-particle resuspension or entrainment (van
Rijn, 1985; Maa and Mehta, 1987). In fact, laboratory
observations as well as field data analysis show that wave
conditions required to generate measurable bottom motion can be
quite moderate compared with conditions required to cause
significant particulate resuspension (Maa, 1986; Suhayda, 1986;
Ross, 1988). Accordingly, the following simple system is
considered.
A two-layered, water-fluid mud system forced by a
progressive, non-breaking surface wave of periodicity specified
by frequency, o, is depicted in Fig. 2. As far as wave dynamics
is concerned we will restrict the problem to one of long waves,
which would therefore be applicable to very shallow coastal or
lacustrine water bodies, or to the margins of deeper ones where
wave action often matters the most. In the case of a rigid
bottom, the shallow water condition is considered to be satisfied
when Ho2/g < 0.1, where H is water depth and g is acceleration
due to gravity. For a given o, this relationship specifies H such
that for shallow water condition to hold, the actual depth must
be equal to or less than that value of H. When the bottom is non-
rigid the maximum water depth to which shallow water condition is
satisfied will be somewhat larger, inasmuch as the wave length
will be greater than in the rigid bottom case.
The upper water layer of thickness H1 and density p, is
considered to be inviscid, which is not unreasonable in
comparison with the highly viscous lower, compliant layer of
fluidized mud considered to be homogeneous and having a thickness
H2, density p2 and dynamic viscosity p. Physical scale arguments
presented by Foda (1989) suggest that viscous dissipation in the
bed may be restricted to a relatively thin boundary layer just
below the mud-water interface. In the present case, however,









energy dissipation is assumed to be distributed over the entire
lower (fluidized mud) layer. Beneath this layer is the bed, which
is assumed to be rigid for the present purposes.
The surficial and interfacial variations about their
respective mean values are qn(x,t) and r2(x,t). The amplitude of
the simple harmonic surface wave is assumed to be small enough to
conform to linear theory, as also the response of the mud layer.
Accordingly, the relevant governing equations of motion and
continuity can be written for the two layers as (Gade, 1958):

Upper layer:

au a1n
at ax- = 0 (1)

au
a u1
at (1-"2) + H1 x = 0 (2)


Lower layer:

au2 an2 anI a2 u
at+ rg ax + (1-r) g x v -2 (3)
az

h au an2
Saxdz + at= 0 (4)
ax at

where u1(x,t) and u2(x,z,t) are the wave velocities, h = H2 + 1'
r = (p2 Pl)/P2 and v = I/p2, is the kinematic viscosity of mud.
Considering the fact that the fluid domain is bounded between z =
0 and H1 + H2, is infinite in extent in the +x direction, the
lower layer is viscous, and the solution sought is harmonic, the
following boundary conditions are imposed:

91(0,t) = a0cosot (5a)

ul(-,t), u2(",z,t), nl(-,t) and r12(-,t)40 (5b)









u2(x,0,t) = 0 (5c)

au2(x,H2,t)/az = 0 (5d)

where ao is the surface wave amplitude at x = 0. Eq. 5a specifies
the surface wave form (progressive, simple harmonic), Eq. 5b
represents the fact that due to viscous dissipation, all motion
must cease at infinite distance, Eq. 5c is the no-slip bottom
boundary condition, and Eq. 5d states that because the upper
layer fluid is inviscid, there can be no stress at the interface.
In order to generalize the solution of Eqs. 1 through 4 and
the boundary conditions (Eq. 5), the following convenient
dimensionless quantities are introduced: ul = ul/oH1, u2 = u2/oH1'
I = l/H1, H2 = 11/H1, H2 = H2/H1, h = H2 + 2", I = ot, k = kH1
(where k is the wave number), x = x/H1, and z = z/H1. Thus Eqs. 1
through 4 become:


Upper layer:

au an^
1 1 1
+ 2-- 0 (6)
aE F ax
r

a 1i
S( 1-2) + = 0 (7)
at ax

Lower layer:

2-
a2 r a2 1-r a1 1 a22
+ 2 2 ~ ~= (8)
at F ax F 2x Re a2
r r


h au x an
ha2 au2
S- d + -= 0 (9)
o ax at


where Fr = o(H1/g)1/2 is the wave Froude number and Re = oH12/v is
the wave Reynolds number. Note that oH1 is the characteristic
shallow water velocity associated with wave motion. Note further









that the dimensionless surface slope term in Eq. 6, as well as
the interfacial and surface slope terms in Eq. 8 are scaled by
1/Fr2, while the dissipation term in Eq. 8 is scaled by 1/Re.
Consider a typical set of characteristic values including o = 1
rad/s, H1 = 1 m, v = 103 m2/s, and r = 0.1. This yields Fr = 0.32
and Re = 103. Thus the coefficient multipliers of the above four
dimensionless terms will be 9.8, 1, 8.8 and 0.001, respectively.
It is thus seen that the multiplier of the dissipation term is
much smaller than those of the surface and interfacial gradient
terms, particularly the former. Yet, of course, dissipation plays
an critical role in the problem in terms of wave damping and a
significant boundary layer effect within the mud. Note also that
at such a low value of the wave Reynolds number, fluid mud motion
is wholly laminar (Maa and Mehta, 1987).
The normalized boundary conditions are expressed as:

r1(0,t) = A cost (10a)


u1(o,t), u2(-o ,z,), (-,~) and 02(c,t) 4 0 (10b)


2 = 0 (10c)

au2/az = 0 (10d)


where A = a0/HI is the normalized wave amplitude. Note that by
virtue of the assumption of rotationality in the lower layer
only, and the shallow water wave condition, only u2 can vary with
z. The solution of Eqs. 6 through 9 with these boundary
conditions is straightforward, and is given in Appendix D.
Results relevant to the experiments conducted are presented
later. While the solution is inherently simplistic, it served as
a useful framework for guiding the interpretation of date which
were obtained via field tower deployment in Lake Okeechobee. This
large and shallow water body in the south-central part of Florida
is well suited to studying wave-mud interaction, as noted in the
next section.









range from 0 to 1.0 m. Notwithstanding evident limitations in
constructing this relationship which, for example, does not
account for changes in wave conditions themselves at a given
water level, this "mean" description does indicate a significant
dependence of the bottom area acted on by the waves at different
water depths. In turn this strong dependence suggests that
seasonal water level variation in this lake is likely to be a
major factor in affecting bottom resuspension and bottom mud
motion characteristics.
During the field deployments, the wave conditions may be
characterized as having been mild, with significant wave heights
on the order of 10 cm or less and periods on the order of 2-3 s.
While such waves did generate horizontal motions within the
bottom mud under close to shallow water conditions, it can be
shown that the maximum bottom stresses would be quite small,
insufficient to cause measurable resuspension (Hwang, 1989).
Therefore, under the given wave conditions, the assumption of
zero resuspension in the theoretical approach was met adequately.
When storm waves do occur, the top ~5 cm thick layer of the
bottom mud tends to dilate due to upward diffusion of sediment to
-10 cm. Above this dilated layer, upward sediment mass transport
tends to be comparatively very small, but the presence of the
dilated layer does tend to complicate the near-bed processes as
far as sediment motion is concerned (Hwang, 1989).
Two additional features of this mud bottom environment are
noteworthy relative to the problem under consideration. Firstly,
the mud throughout includes about 40 % (by weight) material that
is essentially of organic origin and, as a result, the "granular"
density of the composite material is 2.14 g/cm3, which is less
than that for clays for example (~2.65 g/cm3) (Hwang, 1989).
Secondly, the top 5 to 20 cm of the mud has negligible (vane
shear) strength and is believed to be in a fluidized state (Kirby
et al., 1989; Hwang, 1989). This state is brought about partly by
the occasionally significant wave action, but it is believed that
an additional noteworthy factor is the presence of the high
fraction of organic material which, presumably by virtue of









The overall objective of the field investigation was to
record oscillatory motion of fluidized mud in response to wind-
generated waves at a shallow site with a muddy substrate. The
field site and the experiments are noted below.


III. FIELD SITE
The main requirements for the field site were: 1) wave-
dominated environment, 2) shallow water, and 3) a soft muddy
bottom with sizeable thickness of fluidized mud. These conditions
are approximately met in the southeastern part of Lake Okeechobee
close to the shoreline (-1 km offshore) in the vicinity of the
Okeechobee Waterway, a part of the Intracoastal Waterway system.
Fig. 3a shows depths in this rather shallow lake, and
Fig. 3b shows the bottom mud thickness. Depths in Fig. 3a may be
considered to be relative to the top of the mud. The depth datum
is 3.81 m above NGVD, but the actual water level in the lake is
subject to significant variation imposed by the inflows and
outflows which are controlled. Thus, for example, during a field
excursion to collect bottom sediment samples in March, 1988
(Hwang, 1989) the actual water level was about 1.2 m higher than
the datum.
The variation of water level implies the likelihood of a
corresponding variation in the mud bottom area which is
influenced by wave action. Assuming typical storm-induced wave
characteristics and their erodibility potential as represented by
the critical shear stress for erosion, a water depth of 3.4 m can
be calculated (see Appendix A) as the critical depth such that
the bottom will erode only if the actual depth is equal to or
less than this critical depth. Based on this criterion, Fig. 4
shows the lake area influenced by waves at different water levels
(arbitrarily selected to be -1.0 m to +1.5 m relative to datum).
It is observed that when the lake level is less than 0.5 m below
datum, the entire mud bottom area of 528 sq. km is subject to
wave action. On the other hand, when the level is, for example,
1.5 m above the datum, the affected area is reduced to 48 sq. km.
The shape of the curve further implies that the bottom area
influenced by wave action is most sensitive to water level in the









having an open and comparatively strong aggregate structure of
floral origin, prevents rapid dewatering of wave-suspended
surficial deposits even during calms. Hence a bed is not formed
easily in the top layer, although 10 to 20 cm below mud surface,
self-weight does seem to lead to crushing of the aggregates and
consolidation of the deposit. The outcome is a comparatively
uniform density below the top fluidized layer.


IV. EXPERIMENTS
In order to achieve the study objective it was necessary to
obtain the time-series of water level, the corresponding wave
orbital velocities in the water column and induced orbital
velocities in the mud. Water level was measure with a subsurface
mounted pressure gage (Transmetrics, Model P21LA). Water motion
was measured by an electromagnetic (EM) meter (Marsch-McBirney,
Model 521). The EM meter has been used previously with a
reasonable degree of success in the fluid mud environment to
measure tide-induced flows having sediment concentrations up to
about 400 g/l, i.e. 1.27 g/cm3 (Kendrick and Derbyshire, 1985).
In this study however it was decided to measure wave-induced
accelerations instead, in order to obviate likely problems in
interpreting EM meter data in the presence of high concentration
sediment. The use of accelerometer for such a purpose has been
reported previously (Tubman and Suhayda, 1976). A biaxial
accelerometer was used in the present study (Entran, Model EGA2-
C-5DY).
Two field tests were carried out at the selected nearshore
site (Fig. 3a) using a tower shown in Figs. 5 and 6. The site was
in the proximity of the "Green 17" channel marker. At this site,
often prevalent westerly and northwesterly winds generate
suitable waves over a comparatively long fetch. The first test
(test 1) was on December 20-21, 1989, and the second (test 2) on
March 28, 1990. The duration of test 1 was from 1700 hr on
December 20 to 0100 hr on December 21. The duration of test 2 was
from 1730 hr to 2130 hr on March 28.









The aluminum field tower frame assembly was designed for
providing bottom stability and to hold a 4.2 cm diameter aluminum
shaft within a concentric pipe of 5.8 cm o.d. as shown. The tower
had a total height of 2.45 m. The size.at the base was 1.5 m by
1.0 m, tapering to 0.25 m by 0.15 m at the top. The slanted
members were braced together in order to give adequate strength
to the tower against buckling and torsion during installation and
retrieval operations. A wooden plank (base) of 0.8 m by 0.8 m
size was firmly fixed at the top of the tower for mounting the
data acquisition equipment. The tips (pins) of all the four legs
of the tower were made conical so that they could easily
penetrate the soft mud layer and provide stability. Horizontal
braces were provided at the bottom of the tower at an elevation
of 8 cm above the ends of legs. In addition to providing strength
to the tower, these braces arrested the excessive downward
movement of the tower, bringing it to rest over relatively hard
bottom.
At the lower end of the central shaft a holder was provided
to carry the accelerometer mounted in a plexiglass "boat",
consisting essentially of a horizontal oval disc with vertical
guide vanes (see Appendix B). With the accelerometer embedded in
the disc, the boat was made neutrally buoyant at a density of
about 1.07 g/cm3. This arrangement allowed the accelerometer to
be loosely suspended at a desired elevation below the shaft,
constrained only by the vertical play of the shaft in a fluid mud
of 1.07 g/cm3 density. The accelerometer could be rotated by
rotating the shaft itself to orient the device in the desired
direction.
The pressure gage and the EM meter were clamped on to the
concentric pipe (Fig. 7). In test 1 the EM meter was oriented in
such a way as to allow it to record the two horizontal components
of the wave velocities, uI and v1. In test 2, ul and the vertical
component, wl, were measured. The accelerometer was mounted in
such a way as to enable it to record the horizontal component of
acceleration in the dominant wave direction, u2, and the vertical









having an open and comparatively strong aggregate structure of
floral origin, prevents rapid dewatering of wave-suspended
surficial deposits even during calms. Hence a bed is not formed
easily in the top layer, although 10 to 20 cm below mud surface,
self-weight does seem to lead to crushing of the aggregates and
consolidation of the deposit. The outcome is a comparatively
uniform density below the top fluidized layer.


IV. EXPERIMENTS
In order to achieve the study objective it was necessary to
obtain the time-series of water level, the corresponding wave
orbital velocities in the water column and induced orbital
velocities in the mud. Water level was measure with a subsurface
mounted pressure gage (Transmetrics, Model P21LA). Water motion
was measured by an electromagnetic (EM) meter (Marsch-McBirney,
Model 521). The EM meter has been used previously with a
reasonable degree of success in the fluid mud environment to
measure tide-induced flows having sediment concentrations up to
about 400 g/l, i.e. 1.27 g/cm3 (Kendrick and Derbyshire, 1985).
In this study however it was decided to measure wave-induced
accelerations instead, in order to obviate likely problems in
interpreting EM meter data in the presence of high concentration
sediment. The use of accelerometer for such a purpose has been
reported previously (Tubman and Suhayda, 1976). A biaxial
accelerometer was used in the present study (Entran, Model EGA2-
C-5DY).
Two field tests were carried out at the selected nearshore
site (Fig. 3a) using a tower shown in Figs. 5 and 6. The site was
in the proximity of the "Green 17" channel marker. At this site,
often prevalent westerly and northwesterly winds generate
suitable waves over a comparatively long fetch. The first test
(test 1) was on December 20-21, 1989, and the second (test 2) on
March 28, 1990. The duration of test 1 was from 1700 hr on
December 20 to 0100 hr on December 21. The duration of test 2 was
from 1730 hr to 2130 hr on March 28.










component, v2. Data bursting for all the three transducers was at
the rate of 4 Hz for 10 min every hour in test 1, and 5 min every
1/2 hour in test 2. This digitization frequency and record
lengths may be considered to be minimally adequate based on
previous studies (see Mehta and Dyer, 1990). The transducers were
connected to a data acquisition system (Tattletale, Model 6)
mounted on the wooden plank (Fig. 8).
Mean water depth, mud thickness and the depths below still
water level at which the pressure sensor, the EM meter and the
accelerometer were deployed in the two tests are given in
Table 1.


Table 1. Test parameters (depths and elevations)

Test Water Mud Depth below still water level (m)
No. depth thickness
(m) (m) Pressure Velocities Accelerations

1 1.43 0.55 0.54 0.87 1.63
2 1.64 0.35 0.58 1.22 1.69


V. BOTTOM MUD CHARACTERISTICS
In consonance with the nature of the problem and the two-
layered formulation shown in Fig. 2, the bulk density and the
dynamic viscosity can be considered to be the two important
parameters characterizing the mud bottom. Vertical density
profiles in the mud were obtained by a simple bottom coring
procedure (Srivastava, 1983), using a hand-held corer that
yielded approximate variation of density with depth below the
mud-water interface. It should be noted that in the lake
environment this interface is quite well defined during calm
conditions (Fig. 9).
Mud viscosity was measured in a laboratory viscometer at
different mud bulk densities (see Appendix C). The relationship
between the dynamic viscosity and mud density shown in Fig. 10









(see also Fig. C.6 in Appendix C) will be considered to be
adequate in characterizing the wave energy dissipation property
of the mud at different densities. Note that due to limitations
in the apparatus, mud suspensions of densities higher than 1.12
g/cm3 could not be tested. It was therefore assumed that at
higher (up to 1.18 g/cm3) densities, the viscosity could be
obtained by linearly extrapolating the curve shown in Fig. 10.
Mud bulk density profiles (one from test 1 and two from test
2) are shown in Figs. lla,b. The substrate underneath the mud
layer may be considered as "hard"; the transition to hardness
being here defined as the level at which the field tower rested
on its own account, rather than in terms of hardness related to
bottom composition. The mud layer thickness shown in the figures
and given in Table 1 is based on this consideration.
It is interesting to note that in the second test the tower
seemingly rested on a hard, approximately 7 cm thick "lens", with
softer material both above and below this thin lens. The
occurrence of such a lens can be due to peculiarities associated
with episodic accumulation arising from locally resuspended and
allothegenous sediment. Also shown in the figures is the level at
which the accelerometer (AC) was embedded. In test 1 it was 0.2 m
below the mud-water interface and in test 2 it was 0.05 m below
the interface. The corresponding densities were 1.18 and 1.15
g/cm3. Considering that the boat with the accelerometer was
neutrally buoyant at 1.07 g/cm3, the placement of the device at a
somewhat higher density imparted buoyancy which was undesirable.
The problem occurred because of the difficulty in determining mud
density in situ when the acceleometer was deployed.
Mechanistically the problem is somewhat analogous to the wave-
induced motion of a submerged buoy tethered by a rope to the
bottom, with the rope held taut by the buoyancy of the buoy. It
can be shown easily that given physical parameters relevant to
the present problem, the effect of buoyancy on measured
accelerations would be minor.









VI. WATER AND MUD MOTIONS
The time-series of pressure, water velocities and mud
accelerations were analyzed in terms of their spectral properties
and central tendencies. Water pressures were converted to wave
heights via the pressure response factor based on the linear
theory. For a description of the linear theory and wave spectra
see Dean and Dalrymple (1984). In what follows, data from test 1
are discussed, followed briefly by those from test 2.
With reference to test 1, Fig. 12a shows the variation of
the significant wave height, H1/3, with time over the seven hour
test duration (data from the first data block at 1700 hr were
found to be spurious due to lack of adequate time for electronic
system warm-up, and therefore are not included). Zero hour
corresponds to 1800 hr on December 20, 1989. Note that each
hourly data point represents a 10 min average value; 10 min being
the record length for each hourly data block. Under gentle to
moderate breeze, H1/3, is observed to have been rather small,
peaking to 10 cm at 3 hr. In Fig. 12b the corresponding variation
of the dominant (modal) surface wave frequency, f,, defined as
the frequency at the peak of the wave energy density spectrum, is
shown. An example of the spectrum itself is shown in Fig. 13a.
This and all other spectra represent ensemble averages obtained
by selecting a band width of 10 sampling points, the sampling
interval being 0.25 s. The modal frequency variation is compared
with the same determined from the water velocity spectrum (see
for example Fig. 13b). As observed the dominant wind-wave
frequency was comparatively constant, varying between 0.38 and
0.50 Hz, with a mean value of 0.42 Hz over the test duration.
The relative constancy of the wave frequency throughout test
duration suggests that the wind fetch was likely to have been
more or less constant, notwithstanding the fact that the wave
height did vary somewhat more significantly than frequency. In
Fig. 14 the angular direction of the horizontal water velocity
(resultant of the two measured components, u and v) relative to
an arbitrarily selected coordinate (direction) is plotted. This
plot does indicate a comparatively constant direction of wave









approach throughout the test (the angle varied between 350 and
47O, with a mean of 420). The direction was approximately
westerly. This direction corresponds to a lake fetch on the order
of 50 km with a mean depth of around 3 m (Fig. 3a). Selecting a
wave period of 2.5 s corresponds to a wind of 20 km/hr (moderate
breeze), using shallow water forecasting curves for wave
generation over a rigid bottom (Coastal Engineering Research
Center, 1977). However, the forecasted wave height under these
conditions is 40 cm. It can be surmised that, notwithstanding the
approximations (e.g. constant water depth) involved in these
calculations, the discrepancy is likely to be due to significant
wave damping over the mud bottom which stretches over 30 km, so
that only the first 20 km distance can be considered to be over a
rigid bottom.
In Fig. 15, the variance of the resultant velocity
(amplitude) is plotted. Comparing this observed time-variation
with the corresponding variation of wave height in Fig. 12a shows
expected similarities in time-trends.
In Fig. 16a the variances of the horizontal and vertical
components of mud acceleration are plotted over the duration of
test 1. These indicate the latter to be expectedly smaller in
comparison with the former a part of the time, but during the
early (except at 0 hr) and later phases of the test their
magnitudes were generally of the same order. This implies that
the shallow water condition was not quite met, and that wave
orbital motion in the mud varied from practically horizontal
(e.g. at 0 hr when the vertical acceleration was negligibly
small) to circular (e.g. at 1 and 5 hrs). In fact, it can be
easily shown that the shallow water condition was only
appropriate for waves having frequencies less than around 0.2 Hz.
In Fig. 16b the modal frequencies corresponding to the
spectral peaks arising from wind wave action are plotted. An
example of the horizontal acceleration spectrum itself is given
in Fig. 16c. Note that the spectrum shows a marked peak at a very
low frequency corresponding to a long period oscillation distinct
from direct wind forcing. Commensurate peaks also appear in the









wave and velocity spectra of Figs. 13a,b. Vertical acceleration
measurements did not exhibit these low frequency, long wave
signatures for evident reasons, and the corresponding spectra are
not considered further in what follows. The long wave signature
is further discussed later.
Note that the modal frequencies in Fig. 16b correspond to
the portion of the horizontal acceleration spectrum exclusive of
the low frequency signature. The modal frequencies of vertical
acceleration generally coincided (range: 0.42 to 0.48 Hz; mean =
0.40 Hz), given the limits of likely errors due measurement and
analysis procedures, with those of the surface wave and water
velocity given in Fig. 12b, while those of the horizontal
acceleration are observed to be slightly higher (range: 0.40 to
0.62 Hz; mean = 0.51 Hz). This shift may reflect the fact that
for the higher frequencies encountered the shallow water
assumption did not quite hold, and as a consequence there was
greater damping at lower frequencies than at the higher ones. It
is also possible to attribute this frequency shift to likely
limitations in the measurement of accelerations with the "boat",
noting that difficulties with the laboratory setup precluded a
detailed investigation of the motion of the boat in the range of
the very small amplitude motion that was encountered within the
mud during the field tests.


VII. RESPONSE OF LINEARIZED FLUID MUD-WATER SYSTEM
Given the surface wave spectrum such as in Fig. 13a, which
characterizes the measured wave amplitude (A) variation with wave
frequency, Eq. D-15c can be used to calculate the water velocity,
and the time-derivative of Eq. D-15d to calculate mud
acceleration.
An evident difficulty in adapting the simple model to
measurement involves the selection of a representative (constant)
mud density, since as seen from Figs. lla,b the density
characteristically increased quite rapidly with depth below the
interface in the top -10 cm. The matter of selecting mud density
of course bears critically on the rate of energy dissipation, due
to the rather drastic dependence of the mud viscosity on density









(Fig. 10). In order to account for this problem, Maa and Mehta
(1987)and others (e.g. Shibayama et al.,1989) have developed
layered bed models in which the mud density and viscosity can
vary arbitrarily with depth. It was felt that the use of such
models for the present case would essentially amount to an over-
specification of the physical system, whose understanding was
constrained by the rather limited data collection effort.
Furthermore, from the perspective of lake dynamics the important
point to be made for the present case is to show that, even under
relatively weak wave action, mud down to a significant depth
below the interface moves in the shallow parts of the lake,
thereby presumably influencing constituent (e.g. nutrient) fluxes
in a measurable way. In that context, the purpose of the simple
model used here may be considered to be for corroborating the
observation of mud motion through basic physical principles.
For calculating the velocity and acceleration spectra from
the wave energy spectra, the following transfer functions based
on Eqs. D-15a, D-15c and D-15d were used:


From wave energy spectrum to water velocity spectrum:

SU = K S (lla)
ulu, ul rj

where

K = Real(y] (lib)
1 F
r


From wave energy spectrum to mud acceleration spectrum:


Sd2 = K2S (12a)
22a 2

where

2
K. = Real(-igk(1 rk2)[l cosh(mH1z) + sinh(mHlz)
2 1 F 2
r

*tanh(1-i)x]) (12b)









wave and velocity spectra of Figs. 13a,b. Vertical acceleration
measurements did not exhibit these low frequency, long wave
signatures for evident reasons, and the corresponding spectra are
not considered further in what follows. The long wave signature
is further discussed later.
Note that the modal frequencies in Fig. 16b correspond to
the portion of the horizontal acceleration spectrum exclusive of
the low frequency signature. The modal frequencies of vertical
acceleration generally coincided (range: 0.42 to 0.48 Hz; mean =
0.40 Hz), given the limits of likely errors due measurement and
analysis procedures, with those of the surface wave and water
velocity given in Fig. 12b, while those of the horizontal
acceleration are observed to be slightly higher (range: 0.40 to
0.62 Hz; mean = 0.51 Hz). This shift may reflect the fact that
for the higher frequencies encountered the shallow water
assumption did not quite hold, and as a consequence there was
greater damping at lower frequencies than at the higher ones. It
is also possible to attribute this frequency shift to likely
limitations in the measurement of accelerations with the "boat",
noting that difficulties with the laboratory setup precluded a
detailed investigation of the motion of the boat in the range of
the very small amplitude motion that was encountered within the
mud during the field tests.


VII. RESPONSE OF LINEARIZED FLUID MUD-WATER SYSTEM
Given the surface wave spectrum such as in Fig. 13a, which
characterizes the measured wave amplitude (A) variation with wave
frequency, Eq. D-15c can be used to calculate the water velocity,
and the time-derivative of Eq. D-15d to calculate mud
acceleration.
An evident difficulty in adapting the simple model to
measurement involves the selection of a representative (constant)
mud density, since as seen from Figs. lla,b the density
characteristically increased quite rapidly with depth below the
interface in the top -10 cm. The matter of selecting mud density
of course bears critically on the rate of energy dissipation, due
to the rather drastic dependence of the mud viscosity on density









Model application can be illustrated by considering data
from test 1. Water depth H1 is given in Table 1. The choice of
appropriate mud thickness, H2, was obviously difficult in the
same sense as choosing a representative mud density. With regard
to density, a value of 1.18 g/cm3 was selected, corresponding to
the position (AC) of the accelerometer shown in Fig. 11a. The
computations were found to be particularly sensitive to the
choice of H2 by virtue of the effect of X (see Appendix D) on the
kinematics, which in turn signifies the interrelationship between
the mud boundary layer and energy dissipation. Furthermore, as a
rule mud becomes rather immobile at densities exceeding around
1.2 g/cm Therefore, it would be unreasonable to select the
entire mud thickness of 0.55 m for computational purposes. It was
decided to select H2 = 0.283 m (corresponding to a density of
1.22 g/cm3). This selection seemed to give the best results by
way of agreement with data, for the cases examined.
Given the surface wave data represented in Fig. 13a, the
calculated (simulated) and measured water velocity and mud
horizontal acceleration spectra are shown in Figs. 17a,b. Note
that the velocity measurement was 0.87 m below the still water
level (Table 1), an elevation that was likely to have been above
the wave boundary layer, since the total water depth was 1.43 m;
hence the interface was well below the position of the current
meter. Therefore, the model assumption of inviscid upper layer,
when comparing simulated values with data essentially in the
upper portion of the water column, may not be overly limiting.
Data and simulation generally show the same trends. Note that by
virtue of the assumptions of linearity and inviscid upper layer,
the calculated velocity spectrum is closely self-similar to the
wave energy spectrum. Since in the field the measurement of
pressure under non-breaking waves is usually more reliable than
water velocity, the general similarity between the measured wave
energy and water velocity spectra, as well as the general
agreement between the simulated and measured velocity spectra,
can be considered, in a sense, to attest to the reliability of
the water velocity data from the field.









Comparing the simulated and measured horizontal mud
accelerations in Fig. 17b indicates that while the model
generally agrees with the data, the latter show the occurrence of
accelerations at high frequencies which can not be accounted for
by the shallow water model. As noted before, these high frequency
waves were proportionately less damped most probably due to these
being outside the shallow water domain, in intermediate depth.
This type of selective damping of shallow water wave components
relative to those in intermediate depth has been quite well
documented, for example, by Wells and Kemp (1986) in their study
on wave propagation over a large mud flat off Surinam.
In Figs. 18a,b,c measurements and model calculations are
shown for the data block at 5 hr, similar to those for the block
at 1. Once again the interpretations relative to the data, as
well as comparison between data and model calculations, remain
the same. All measured spectra (wave energy, water velocity and
horizontal mud acceleration) for test 1 are given in Appendix E.
With reference to mud acceleration spectra, seemingly exaggerated
peaks corresponding to the long wave spectral signal occasionally
appeared, e.g. at 0, 2 and 6 hrs in test 1. We are uncertain
about the cause of this type of a response; it is not clear for
instance if this was due to a problem in the acceleration
measurement, or if there was a physical cause. Granting no data
error, the large peak would mean much less damping of the low
frequency than that of the wind wave. While this implication
seems to be borne out by theory as noted later, the implied
dependence of the damping coefficient on wave frequency is
greater than what the theory indicates. A further look at this
phenomenon in future will be necessary.
Data from test 2 were found to be somewhat unsatisfactory in
the sense that wave action was very weak throughout the test
duration, with the result that the wind induced wave spectra
contained very little energy. For illustrative purposes, spectra
obtained from the first block (at 1800 hr on March 28, 1990) are
shown in Figs. E.7 through E.11.
The low energy content of wind waves during the second test
is evidenced by a comparison, for example, between Figs. 13a and









E.7. Somewhat surprisingly, however, the low frequency peak is
quite prominent (relative to wind wave peak) in Fig. E.7. We
further note that: 1) The forcing wave and the low frequency
occur at about the same frequencies (0.57 Hz and 0.05 Hz) as in
test 1, which in turn suggests that the general nature of the
phenomena at the test site are likely to be persistent. (Note
that test 1 was conducted at the inception of the winter period
while test 2 was at the end of this period.) In other words, the
conclusions derived from the examination of test 1 data are
likely to be applicable to events in the study area on more than
a mere isolated basis. 2) The persistence of the low frequency
peak, even in the absence of significant wind energy, is
noteworthy in that the occurrence of this peak is somewhat
enigmatic insofar as the causative mechanism is concerned.
With regard to the water velocity spectra is Figs. E.8 and
E.9, recall that the first is the horizontal component and the
second vertical. Thus the horizontal component may not precisely
reflect the velocity in the wave direction (which, unlike in
test 1, could not be determined). On the other hand, the general
features in Fig. E.8 are commensurate with those in Fig. E.7.
Here again a strong low frequency, long wave signature is
observed. Vertical velocities (Fig. E.9) were expectedly smaller
and did not show the low frequency signature.
Comparing Figs. E.10 and E.11 for the horizontal and
vertical mud accelerations it is seen that the wind-induced
signature was too small to be recorded. Furthermore, the
horizontal acceleration was more important than vertical at the
low frequency.
An interesting application of the model relates to wave
damping in the lake. Thus, for example, at a frequency of 0.4 Hz,
the wave damping coefficient calculated by the model is 0.0034
m-'. Over rigid beds on the other hand, shallow water waves
attenuate much more slowly; the damping coefficients are only on
the order of 10-5 m-1 (Ippen and Harleman, 1966). Thus, wind
generated waves will be smaller over a compliant bottom. As
noted, during the experiment, waves at the site were only a









quarter as high as those that would be generated over a rigid
bed.


VIII. LOW FREQUENCY SIGNATURE
The dominant frequency of the low frequency spectral
signature is plotted in Fig. 19 for test 1 from wave energy,
water velocity and horizontal mud acceleration spectra. It is
observed that all the spectra yield rather consistent values
ranging from 0.029 Hz to 0.049 Hz, with a mean of 0.043 Hz. In
practical terms the frequency was seemingly unaffected by changes
in the wave conditions during the test. Referring to Figs. 13a,b
and 16c (data block at 1 hr, test 1), it is seen that the low
frequency peak relative to the forcing wave (modal) peak was
enhanced in the mud (Fig. 16c) relative to that in water (Figs.
13a,b). Model results for the same data block in Figs. 17a,b show
similar trends, and indicate that this type of a response for
shallow water waves is due to frequency dependent wave damping as
can be gleaned from the attenuation coefficient characterized in
Fig. D.2. Thus for example, selecting 0.4 Hz and 0.04 Hz as
representative frequencies for the forcing wave and the low
frequency, respectively, the corresponding damping coefficients
are 0.0034 m-1 and 0.0013 m-1, which indicates relatively much
less damping at the lower frequency. It is also noteworthy that
by integrating the acceleration twice, the horizontal
displacement of mud can be shown to increase with decreasing
frequency. Maximum displacements on the order of 2 mm can be
shown to have occurred at ~0.03 Hz in both tests.
Some comments on the causative mechanism for the low
frequency signature are in order. In that context it must be
noted that the dominant period of seiching in Lake Okeechobee is
in the range of 5 to 6 hr (Ahn, 1989), i.e. a frequency on the
order of 10- Hz, which is considerably smaller than the observed
low frequency peak. It seems plausible that the low frequency
wave is in fact a second order effect resulting from wind induced
wave forcing leading to surf beat. This effect, in which short
period wind-waves are modulated by a longer period wave of very









low amplitude, commonly observed along the open coast having a
hard or sandy bottom with mild slopes and having a frequency
range of 3x10-3 to 8x10-3 Hz (Wiegel, 1964), may be modified by a
compliant bottom as in the present case, but it serves the
purpose to examine this mechanism assuming a rigid bottom, at
least for arriving at a qualitative explanation for the
occurrence of the long wave signature.
We consider two waves, rl(0,t) and r2(0,t), of respective
frequencies al and 02 = 01 + Ao, where Ao is a small difference:


1 = alcos(olt E1) (13a)

T2 = a2cos(o2t E2) (13b)

Here, es and E2 are the phase lags. Assuming further that the
amplitude a = al = a2, the resultant (forcing) wave, nr = 1 + n2
can be shown to be


r = 2acos(ot s)cos(A't2 As) (14)


where o = (al + 02)/2, e = (E + E2)/2, and As = el E2. It is
thus seen that the sinusoidal wave term, 2acos(ot e), is
modulated by cos[(Ao.t-As)/2], which causes the well known "beat
effect" due to wave groupiness.
The fact that real waves have finite amplitudes means that
higher (than first) order effects arising from changes in water
surface elevation and associated kinematics cannot always be
ignored. An effect germane to the present case is the setting up
of a long wave which modulates the wave given by Eq. 14. This
forced long wave follows from inclusion of the kinetic head term
in the dynamic free surface boundary condition (DFSBC), and
evaluation of the mean (relative to 0) water surface profile by
carrying out the computations to second order (Longuet-Higgins
and Stewart, 1962; Dean and Dalrymple, 1984). Without restricting
the problem to shallow water, the wave-averaged DFSBC is:










1n (u2 + w2) + (
2g g azat (15)


where the overbar represents short period (2n/o) averaging, u and
w are the horizontal and vertical velocity components, and 4, the
well known form of the potential function which satisfies the
boundary value problem (assuming a rigid bottom for this
simplified case), is given by

S_ ag cosh k(H + z) sinot (16)
o cosh kH

where H is the water depth. Noting that u = a8/ax and w = -
a)/az, it can be easily shown that

a- 2k
= sinh kH [1 + cos(Ao.t E)] (17)


where k = (kI + k2)/2. The first term on the right hand side
represents a steady set down, and the second is the forced long
wave. The energy spectrum (or the energy density spectrum) thus
obtained is shown in Fig. 20, which shows that the long wave,
corresponding to two forcing waves at frequencies fl = oa/2nr and

f2 = 02/2Tr and amplitude a, has a frequency ft = Ao/2i and
amplitude a2k/sinh kH. It is important to note that, comparing
Eq. 17 with Eq. 14 indicates that the long wave is i radians
(1800) out of phase with the wave group envelope of the forcing
wave.
When forcing is represented by a continuous spectrum, the
treatment for determining the forced long wave spectrum becomes
involved (Sharma and Dean, 1979). Here a very approximate
approach is selected. Consider for example the double peaked
velocity spectrum corresponding to the forcing wave in Fig. 13b.
Assuming the two peak frequencies to be the primary contributors
to the corresponding long wave peak, we have the following
parameters (obtained from the Fourier series of the corresponding
surface wave record): a, = 2.4 cm, a2 = 2.6 cm, long wave









amplitude aZ = 0.30 cm, fl = 0.38 Hz, f2 = 0.42 Hz, fZ = 0.049
Hz, E1 = 1910, 62 = 1710, and Ea = -650. Note that Af = f2 fl
0.04, which is reasonably close to fZ. The forcing wave (nr) and
the forced wave, ng, are plotted against p = Ao.t-As in Fig. 21.
It is evident that the forced wave is almost i radians out of
phase with the short period wave envelope as the theory would
require. If we assume shallow water condition, the forced wave
amplitude (from Eq. 17) would be a2/H. Given a = 0.5(al + a2) =
2.50 cm and H = 143 cm, a2/H = 0.044 cm. Thus the theory
underpredicts the amplitude significantly, which might be due to
inherent theoretical limitations as well as the rather gross
assumptions made in applying the theory to the present case.


IX. CONCLUDING REMARKS
Notwithstanding the limited nature of the data obtained, the
complexities in modeling mud motion and the rather obvious
constraint in simulation arising from the shallow water
assumption as well as others (e.g. inviscid water layer,
linearized response, particularly of the mud layer), it is seen
from the data and simulation that measurable mud accelerations
can occur tens of centimeter below the mud-water interface, under
wave action that is mild enough to preclude any measurable
erosion of the interface. It can be easily shown that
corresponding maximum horizontal displacements on the order of 2
mm occurred at the low frequency end (~0.03 Hz) of the spectrum
in both tests. The displacement spectrum, Sdd, is obtained from
the wave spectrum, Sp, as follows:

Sdd = Kd2Sn (18a)

where

kH rk 2H2
K =Real(i (1 21 )[1 cosh(m] +
F F 1
r r

mH2
tanh(--2) sinh(2)]} (18b)
1 1









quarter as high as those that would be generated over a rigid
bed.


VIII. LOW FREQUENCY SIGNATURE
The dominant frequency of the low frequency spectral
signature is plotted in Fig. 19 for test 1 from wave energy,
water velocity and horizontal mud acceleration spectra. It is
observed that all the spectra yield rather consistent values
ranging from 0.029 Hz to 0.049 Hz, with a mean of 0.043 Hz. In
practical terms the frequency was seemingly unaffected by changes
in the wave conditions during the test. Referring to Figs. 13a,b
and 16c (data block at 1 hr, test 1), it is seen that the low
frequency peak relative to the forcing wave (modal) peak was
enhanced in the mud (Fig. 16c) relative to that in water (Figs.
13a,b). Model results for the same data block in Figs. 17a,b show
similar trends, and indicate that this type of a response for
shallow water waves is due to frequency dependent wave damping as
can be gleaned from the attenuation coefficient characterized in
Fig. D.2. Thus for example, selecting 0.4 Hz and 0.04 Hz as
representative frequencies for the forcing wave and the low
frequency, respectively, the corresponding damping coefficients
are 0.0034 m-1 and 0.0013 m-1, which indicates relatively much
less damping at the lower frequency. It is also noteworthy that
by integrating the acceleration twice, the horizontal
displacement of mud can be shown to increase with decreasing
frequency. Maximum displacements on the order of 2 mm can be
shown to have occurred at ~0.03 Hz in both tests.
Some comments on the causative mechanism for the low
frequency signature are in order. In that context it must be
noted that the dominant period of seiching in Lake Okeechobee is
in the range of 5 to 6 hr (Ahn, 1989), i.e. a frequency on the
order of 10- Hz, which is considerably smaller than the observed
low frequency peak. It seems plausible that the low frequency
wave is in fact a second order effect resulting from wind induced
wave forcing leading to surf beat. This effect, in which short
period wind-waves are modulated by a longer period wave of very









Although bioturbation does not seem to be a significant
factor in Lake Okeechobee (Kirby et al., 1989), the effect of
persistent mud oscillation, even though very small, can be
germane to likely changes in the rates of exchange of phosphorus
and other water quality influencing chemical constituents.
Similarly, there may be an effect on the formation and upward
transport of gas bubbles which occur abundantly in the muddy area
of the lake (Kirby et al., 1989). It is believed that gas bubbles
contribute measurably to nutrient dynamics in the lake.
The "openness" of the particulate matrix of the bottom mud
in this lake seems to be greatly controlled by the presence of a
significant fraction of floral organic matter, so that persistent
mud motion may effectively increase pore water transport, and
associated constituent exchange. While this issue does not quite
fall within the rubric of the present scope of work, we recommend
that it be examined in the light of these findings.
We wish to point out a further issue that is related to the
constitutive behavior of bottom mud. While in this study we
considered mud up to a density of about 1.2 g/cm3 to be a highly
viscous fluid, careful measurements (e.g. Sills and Elder, 1981)
indicate that, at least under quiescent conditions, at densities
exceeding about 1.1 g/cm clayey mud typically exists as porous
solid rather than a fluid. The organics-rich Lake Okeechobee mud
is prone to remain in a fluidized state at densities at least up
to about 1.065 g/cm3 (Hwang, 1989). It is therefore unclear if at
1.18 g/cm3 density this mud is normally (i.e. in the absence of
episodic wave action) fluidized. Hence the chosen model
description may be approximate in this respect. Nonetheless,
since the model calculated mud accelerations compare favorably
with the measured ones at least in order of magnitude, the
description of mud as a fluid seems acceptable, although a better
physical description, which recognizes the transition from the
fluid phase to the solid phase, would indeed constitute a
worthwhile improvement.









amplitude aZ = 0.30 cm, fl = 0.38 Hz, f2 = 0.42 Hz, fZ = 0.049
Hz, E1 = 1910, 62 = 1710, and Ea = -650. Note that Af = f2 fl
0.04, which is reasonably close to fZ. The forcing wave (nr) and
the forced wave, ng, are plotted against p = Ao.t-As in Fig. 21.
It is evident that the forced wave is almost i radians out of
phase with the short period wave envelope as the theory would
require. If we assume shallow water condition, the forced wave
amplitude (from Eq. 17) would be a2/H. Given a = 0.5(al + a2) =
2.50 cm and H = 143 cm, a2/H = 0.044 cm. Thus the theory
underpredicts the amplitude significantly, which might be due to
inherent theoretical limitations as well as the rather gross
assumptions made in applying the theory to the present case.


IX. CONCLUDING REMARKS
Notwithstanding the limited nature of the data obtained, the
complexities in modeling mud motion and the rather obvious
constraint in simulation arising from the shallow water
assumption as well as others (e.g. inviscid water layer,
linearized response, particularly of the mud layer), it is seen
from the data and simulation that measurable mud accelerations
can occur tens of centimeter below the mud-water interface, under
wave action that is mild enough to preclude any measurable
erosion of the interface. It can be easily shown that
corresponding maximum horizontal displacements on the order of 2
mm occurred at the low frequency end (~0.03 Hz) of the spectrum
in both tests. The displacement spectrum, Sdd, is obtained from
the wave spectrum, Sp, as follows:

Sdd = Kd2Sn (18a)

where

kH rk 2H2
K =Real(i (1 21 )[1 cosh(m] +
F F 1
r r

mH2
tanh(--2) sinh(2)]} (18b)
1 1









X. REFERENCES


Ahn K. (1989). Wind-wave hindcasting and estimation of bottom
shear stress in Lake Okeechobee. M.S. Thesis, Univ. of Florida,
Gainesville.

Coastal Engineering Research Center (1977). Shore Protection
Manual, Vol. 1, U.S. Army Engineer Waterways Experiment Station,
Vicksburg, MS.

Dean R.G. and Dalrymple R.A. (1984). Water Wave Mechanics for
Engineers and Scientists. Prentice-Hall, Englewood Cliffs, NJ.

Dixit J.G. (1982). Resuspension potential of deposited kaolinite
beds. M.S. thesis, Univ. of Florida, Gainesville.

Foda M.A. (1989). Sideband damping of water waves over a soft
bed. Journal of Fluid Mechanics, 201, 189-201.

Gade H.G. (1958). Effects of non-rigid, impermeable bottom on
plane surface waves in shallow water. Journal of Marine Research,
16(2), 61-82.

Hwang K.-N. (1989). Erodibility of fine sediment in wave-
dominated environments. Rept. UFL/COEL-89/017, Coastal and
Oceanographic Engineering Dept., Univ. of Florida, Gainesville.

Ippen A.T. and Harleman D.R.F. (1966). Tidal dynamics in
estuaries. In: A.T. Ippen ed., Estuary and Coastline
Hydrodynamics, McGraw-Hill, New York, 493-545.

Jonsson I.G. (1966). Wave boundary layer and friction factors.
Proceedings of the 10th Coastal Engineering Conference, Vol. 1,
American Society of Civil Engineers, New York, 127-148.

Kendrick M.P. and Derbyshire B.V. (1985). Monitoring of a near-
bed turbid layer. Rept. SR 44, Hydraulics Research, Wallingford,
U.K.

Kirby R.R., Hobbs C.H. and Mehta A.J. (1989). Fine sediment
regime of Lake Okeechobee, Florida. Rept. UFL/COEL-89/009,
Coastal and Oceanographic Engineering Dept., Univ. of Florida,
Gainesville.

Longuet-Higgins M.S. and Stewart R.W. (1962). Radiation stress
and mass transport in gravity waves, with application to 'surf
beat'. Journal of Fluid Mechanics, 13, 481-504.

Maa P.-Y. (1986). Erosion of soft muds by waves. Rept. UFL/COEL-
TR/059, Coastal and Oceanographic Engineering Dept., Univ. of
Florida, Gainesville.

Maa P.-Y. and Mehta A.J. (1987). Mud erosion by waves: a
laboratory study. Continental Shelf Research, 7(11/12), 1269-
1284.









Mehta A.J. (1989). On estuarine cohesive sediment suspension
behavior. Journal of Geophysical Research, 94(C10), 14303-14314.

Mehta, A.J. and Dyer K.R. (1990). Cohesive sediment transport in
estuarine and coastal waters. In: The Sea, Vol. 9: Ocean
Engineering Science, Part B. LeMehaute and D.M. Hanes eds.,
Wiley, New York, 815-839.

Ross M.A. (1988). Vertical structure of estuarine fine sediment
suspensions. Ph.D. Dissertation, Univ. of Florida, Gainesville.

Ross M.A. and Mehta A.J. (1990). Fluidization of soft estuarine
muds by waves. In: The Microstructure of Fine-Grained Sediments:
From Mud to Shale, Ch. 19, R.H. Bennett ed., Springer-Verlag, New
York, 185-191.

Sharma J.N. and Dean R.G. (1979). Development and evaluation of a
procedure for simulating a random directional second order sea
surface and associated wave forces. Ocean Engrg. Rept. No. 20,
Dept. of Civil Engrg., Univ. of Delaware, Newark.

Shibayama T., Aoki T. and Sato S. (1989). Mud mass transport due
to waves: a visco-elastic model. Proceedings of the 23rd Congress
of I.A.H.R., Ottawa, Canada, B567-B574.

Sills G.C. and Elder D. McG. (1986). The transition from sediment
suspension to settling bed. In: Estuarine Cohesive Sediment
Dynamics, A.J. Mehta ed., Springer-Verlag, Berlin, 192-205.

Srivastava M. (1983). Sediment deposition in a coastal marina.
Rept. UFL/COEL/MP-83/1, Coastal and Oceanographic Engineering
Dept., Univ. of Florida, Gainesville.

Suhayda J.N. (1986). Interaction between surface waves and muddy
bottom sediments. In: Estuarine Cohesive Sediment Dynamics, A.J.
Mehta ed., Springer-Verlag, Berlin, 401-428.

Tubman M.W. and Suhayda J.N. (1976). Wave action and bottom
movements in fine sediments. Proceedings of the 15th Coastal
Engineering Conference, Vol. 3, American Society of Civil
Engineers, New York, 1168-1183.

van Rijn L.C. (1985). The effect of waves on kaolinite/sand beds.
Rept. M2060, Delft Hydraulics, Delft, The Netherlands.

Wells J.T. and Kemp G.P. (1986). Interaction of surface waves and
cohesive sediments: field observations and geologic significance.
In: Estuarine Cohesive Sediment Dynamics, A.J. Mehta ed.,
Springer-Verlag, Berlin, 43-65.

Wiegel R.L. (1964). Oceanographical Engineering, Prentice-Hall,
Englewood Cliffs, NJ.









Williams D.J.A. and Williams P.R. (1989a). Rheology of
concentrated cohesive sediments. Journal of Coastal Research,
Special Issue No. 5, 165-173.

Williams P.R. and Williams D.J.A. (1989b). Rheometry for
concentrated cohesive suspensions. Journal of Coastal Research,
Special Issue No. 5, 151-164.









APPENDIX A
INFLUENCE OF WATER LEVEL ON MUD AREA SUBJECT TO RESUSPENSION


The area of mud zone in Lake Okeechobee subject to wave
resuspension was calculated as a function of the lake water
level. A rectangular computational domain was selected including
the entire lake (Fig 3a). The domain was divided into 58 by 64
square grid cells of sides dx = dy = 1 km. Depth values at grid
intersections were obtained from a survey taken in the summer of
1989 (Ahn, 1989). A representative depth for each cell was
computed as the average of the depths at the four intersection
points of the cell.
Mud thickness in the lake is highly variable (Fig. 3b). For
the present purpose the muddy zone was considered to be that zone
having mud thickness in excess of 10 cm, assuming the area over
which mud thickness is between 0 and 10 cm to be too thin and
patchy to contribute measurably to resuspended sediment mass. The
total effective mud area of this lake under this assumption is
528 km2.
The critical erosional depth, H0, over mud bottom in the
lake was considered to be 3.4 m. In other words, that portion of
the mud bottom over which the actual depth at a given water level
is greater than H0, no erosion can be considered to take place.
The value of H0 of course depends on the wave characteristics
(height and period), the bed resistance coefficient and the
critical bed shear stress for erosion. Since in general wave
characteristics vary with wind speed and direction, and with
bottom conditions, a unique value of H. can not be considered to
exist for the lake. In addition, the bottom resistance
coefficient and the critical bed shear stress also depend on the
composition and form of the mud surface. For the present limited
purposes however, it was considered sufficient to select
representative values of wave and bottom related parameters in
order to obtain a single representative value of H0. Therefore, a
wave height of 0.6 m, period of 3 s (Ahn, 1989), Manning's bed
resistance coefficient n = 0.011 corresponding to a smooth









cohesive bed (Dixit, 1982), and a critical bed shear stress of
0.5 N/m2 (Hwang, 1989) were selected. The selected wave height
and period may be considered to represent moderately strong storm
wave conditions in the lake. The value of H. = 3.4 m was derived
using linear wave theory following standard approach (Jonsson,
1966).
The number N of mud bottom cells whose depths were less than
H. was obtained at different water levels in increments of 0.5 m
between +1.5 m and -1.0 m relative to NGVD datum. The total
erosional area of muddy zone was then approximated as N-dx-dy.
Results are given in the following table:


Table A.1 Variation of erodible mud area with relative water
level.

Relative water level (m) Erosional mud area (km2)

-1.0 528
-0.5 528
0.0 471
0.5 237
1.0 97
1.5 48









APPENDIX B
OPERATION OF THE ACCELEROMETER


It was required to calibrate the Entran (Model EGA2-C-5DY)
biaxial accelerometer and to provide a suitable mount for
deploying the device. Calibration was carried out with a static
arrangement followed by a dynamic check, as briefly noted below.
The accelerometer uses a Wheatstone bridge arrangement
consisting of semiconductor strain gages. These gages are bonded
(two on each side) to a simple cantilever beam which is end
loaded with a small mass. Under acceleration, a "g" force is
created by the mass, which in turn causes a bending moment to the
beam. The strain due to the moment results in a bridge imbalance.
With an applied voltage, this imbalance produces a millivolt (mV)
deviation at the bridge output, which is proportional to the
acceleration vector. The x- and y-channels have an acceleration
range from -5g to 5g.
The static calibration procedure consisted of tilting the
accelerometer at different angles in the -90 to 90 degree range
with respect to the vertical, unstrained position (0 degree) of
the mass below the beam. The voltage was set equal to zero in the
unstrained position, and the millivolt output (mV) at other
angles, 8, was measured. The calibration plot consisted of the
relationship between the acceleration component gsine and the
corresponding mV for both the channels. The responses were found
to be linear, and can be described by the equations:


ax = -1.98976 mV + 4.9745 (B-l)

a = 2.0772 mV 5.2401 (B-2)


where ax and ay are the accelerations in m/s2
The dynamic check involved testing the device in a wave
flume in the Coastal Engineering Laboratory. The accelerometer
was embedded in a plexiglass "boat" shown in Fig. B.1. By gluing
a styrofoam block of suitable size underneath the boat together
with the accelerometer, the device was made practically neutrally









buoyant in water, and the boat was tethered between two metal
frame supports via two rather loose strings running horizontally
between the supports so as to allow a free play (heave and surge)
of the boat in the vertical plane. On the other hand, yawing and
rolling were restricted by this arrangement. Since the
longitudinal axis of the boat was set in the direction of the
wave motion, there was not much swaying motion, and the largely
irrotational particle motion under waves did not cause too much
pitching.
The actual test consisted of recording the ax and ay
components of the acceleration of the boat orbiting under wave
motion, simultaneously with the boat's orbital amplitudes
visually, using vertical and horizontal scales mounted on the
glass side wall of the flume (see Dixit, 1982, for a description
of the flume). A series of tests were carried out at different
wave frequencies. The measured horizontal displacement of the
boat orbit was compared with the same quantity obtained by twice
time-integrating the corresponding components of acceleration.
The vertical motions were found to be too small to measure
accurately.
The comparison is shown in Fig. B.2. The deviations from the
450 line may not indicate random error. The response of the boat
may have been dependent on the wave frequency and hence on the
magnitude of water displacement, although any basis for such a
dependence could not be identified. Overall there seems to be an
agreement between measurement and calculation. Hence Eq. B-1 was
considered to be reasonable. The same was assumed to be the case
with respect to Eq. B-2 in the absence of direct evidence (for or
against such an assumption).









APPENDIX C
MEASUREMENT OF MUD VISCOSITY


Mud viscosity was measured using a Brookfield (Model LVT)
viscometer and miniature vanes; see for example Maa (1986) for a
description of the measurement procedure. The measured
relationships between the applied stress and the rate of strain
(shearing rate) are shown in Figs. C.1 through C.5 for five
different densities of the lake mud. Within the range of shearing
rates considered, all the samples indicated a distinctly
pseudoplastic (shear thinning) behavior. For shearing rates
exceeding around 2 to 4 s-1, the linear part of the relationship
implies a constant viscosity. Conveniently referring to this
viscosity as pH at "high" shearing rates, these viscosities
(relative to the viscosity of water at 22 OC) are plotted against
the mud density, P2, in Fig. C.6. Note the two orders of
magnitude increase in the relative viscosity over a small
increase in density from 1 to about 1.02 g/cm3. Over the
subsequent range of density up to 1.1 g/cm3 the increase in
relative viscosity is observed to have been less significant.
Any interpretation of data obtained using vanes, in terms of
a relationship between viscosity and the structure of the
sediment aggregate matrix must be treated with circumspection,
since vanes, by their very presence, break up the sediment matrix
at the cylindrical surface determined by the vane dimensions
(Williams and Williams, 1989b). Furthermore, viscosity is not
characterized by density in a unique sense, insofar as viscosity
and density are independent physical properties of fluids.
Nevertheless the observed trend does seem to suggest a rather
significant influence of sediment packing, as reflected by the
bulk density, on the dynamic viscosity. At densities less than
about 1.02 g/cm3 the aggregate structure rapidly became tightly
packed with increasing density. However, further increase in
density apparently did not drastically alter the compact
arrangement attained at 1.02 g/cm3. Since the "granular" density
of the sediment was 2.14 g/cm3, the sediment concentration









corresponding to 1.02 g/cm3 would be 37.5 g/l. The corresponding
volume fraction would be 0.014. It is conceivable that this
volume fraction approximates the so-called critical volume
fraction above which the rigidity of the aggregate matrix, as
reflected by the shear modulus of elasticity, increases rapidly
(Williams and Williams, 1989a).
Typical rates of shearing in mud are lower than that for
which the viscosity-density relationship at "high" rate of
shearing in Fig. C.6 (Maa and Mehta, 1987). In fact in the
present application the rate of shearing averaged over mud
thickness (considering it to be representative of local rate of
shearing) was well below 1 s-1, closer to 0.1-0.2 s-1. In order to
estimate the viscosity, L', at such low rates of shearing, a
tangent was drawn through the data points at low shearing rates
as illustrated in Fig. C.3. Viscosities calculated in this way
from the tangent slope are also plotted against the corresponding
mud density in Fig. C.6. In general these viscosities are an
order higher in magnitude than PH values at the same value of
density, but show the same general trend with respect to density
change as pH. This similarity in trend reinforces the earlier
surmised influence of density in governing the packing
arrangement of the sediment aggregates.









APPENDIX D
INVISCID-VISCID FLOW PROBLEM SOLUTION

The solution to Eqs. 6 through 9, given boundary conditions
10a through 10d proceeds as follows:
We begin by assuming the following harmonic solutions for
r., ul, q2 and u2 according to:

Ti = A exp[i(kx E)] (D-la)

u = B exp[i(Hx E)] (D-lb)


q2 = C exp[i(Hx E)] (D-1c)

u2 = D'E(z) exp[i(kx E)] (D-ld)

where k = kH1 is the dimensionless wave number, and B, C, D and
E(z) are unknown coefficients representing the amplitudes of ul'
r2 and u2, respectively. The amplitude of u2 is treated as having
a z-independent part, D, and a z-dependent part, E. These
coefficients are evaluated by substitution in Eqs. 6 through 9
together with the boundary conditions, Eq. 10.
From Eqs. 6, D-la, b we obtain


B = A 2 (D-2)
F
r

Eqs. 7, D-la, b, c and D-2 yield

I?2
C = A (1 2 (D-3)
F
r

From Eqs. 8, D-la, b, c, d, D-2, and D-3 we obtain


D (- E" + i E) = iA (1 r -] (D-4)
F F
r r

where E" = 82 /8z2. Next we let










G = iA- 2([l-r k21)
F F
r r

and further let

D = Re G

Hence

E" + iRe E = 1

which can be readily solved to yield

S+ l h(m) + Msinh(m
E = Re + Mlcosh(m )+ M2sinh(mz)


where


1/2
m = (1-i)( -


1/2
i )


Now the conditions 10c and 10d, respectively, yield


i
1 Re


M2 = Mtanh(mH2)


Hence D-8a becomes


= Re [1-cosh(mz) + tanh(mH2).sinh(mz)]


Next, from Eq. 9,


D
C


E d

0


Substituting D-10 into D-ll and carrying out the integration
yields


S D = i_ D i + tanh(mi)[cosh(mh)-1] sin(mE),
S C Re m


(D-12)


(D-5)


(D-6)


(D-7)


(D-8)


(D-9)


(D-9a)


(D-9b)


(D-10)


(D-11)









Next we introduce the linearizing approximation h z H2, since 92
is small. Then, simplifying D-12 and substituting for C and D
from D-3 and D-6 gives


(l-r Z2)
2 H2 F
(2 = H 2 r r (D-13a)
F (12 2 2)
F
r

where


tanh(mi2)
F = 1 (D-13b)
mH2


Solving for k/Fr


1/2
2 1/2
S 1 + H2r + [(1 + H2F) 4rH2r] /(D-
= ([ --] (D-14a)
r 2rH2r

Thus


S f(IH2r,Re) (D-14b)
r


i.e. k depends on H2, r, Re and Fr and, further, the wave number,
k, also varies with water depth H1. The two solutions for the
dimensionless wave number k from Eq. D-14a correspond to the +
and signs; + sign corresponding to a larger amplitude at the
interface relative to that at the surface, and sign
corresponding to the opposite case. We proceed with the latter,
since selection of the former would violate conservation of
energy. For this situation, coefficients B, C and D are obtained
from D-2, D-3, D-5 and D-6; hence i1, ,2' U1 and u2 can now be
written as









i1 = A exp[i(Hx E)] (D-15a)


12 = A[1 (F ]exp[i(kx E)] (D-15b)
r



1 = A -2 exp[i(Rx E)] (D-15c)



f r; 2
u2 = A 2- [l-r(F ) ][1 cosh(mi) + tanh(mffi2)
F r


sinh(mz)]exp[i(kx E)] (D-15d)

As observed 12 is damped relative to q1 by the multiplier
1-(k/Fr)2. Likewise, u2 is damped relative to ul by the two
multipliers of Ak/Fr2 (amplitude of ul) in D-15d.
Next we seek the dispersion relationship from the real part,
kr, of the wave number k, and an expression for the wave
attenuation coefficient k,, the imaginary part of k. Surface wave
attenuation is then specified as ax = a0exp(-k x), where a0 = AH1,
and ax is the amplitude at a distance x. Note that with this
interpretation of A, Eqs. D-15a through D-15d satisfy the
boundary conditions 10b, as the wave amplitudes and corresponding
velocities vanish at infinite distance (from x=0, where ax = ao).
We have, by definition,

S= r + ii. (D-16a)

and let

r 2
Y = i = YR + iY (D-16b)
r
and









H2r = R + ii


(D-16c)


Hence D-14a (with sign) becomes


2 1/2
= 1 + R + iI [(1 + R + ii) 4r(R + il)]
2r(R + il)
From which (D-17)


R 1= i 2 R[1+R-cos(p2+ 2)1/4]+ I[I-sin (p2 +q21/4]}
2r(R + I)

(D-18a)

Y = 1 2[RR[I-sin (p+q2)1/4]- I[1+R-cose (p2 +q2/4
S2r(R2+ I2 2

(D-18b)


where


where


p = (1+R) 4rR I2


q = 2I(1+R 2r)


R = H1 exp(4X) 1 + 2sin(2x)'exp(2x) ]
2 2x[exp(4x) + 1 + 2cos(2x).exp(2x)


I= H[1 exp(4X) 1 2sin(2x)*exp(2x)
2x[exp(4x) + 1 + 2cos(2x)-exp(2x)



1/2 1/2
x Re H 2(2
2 2 ~ H2 )


Then, from D-16a, D-16b, D-18a and D-18b we obtain


= 2(Y + Y 1/2] Y /2
which is the desired dispersion relationship, and
which is the desired dispersion relationship, and


(D-19a)


(D-19b)


(D-19c)



(D-19d)


(D-19e)


(D-20)










S=1 2 1+ /2 (D-21)
Sr R R)(D-21)

which is the desired expression for the wave attenuation
coefficient.

It can be readily shown (see also D-14b) that kr/Fr depends
on H2, r and X. Note that X is mud layer thickness normalized by
(2v/o)1/2, which is twice the thickness of the laminar, wave-
induced (mud) bottom boundary layer (Dean and Dalrymple, 1984).
For a selected value of r = 0.15 (e.g. corresponding to test 1),
this dependence is shown in Fig. D.1 for H2 ranging from 0.1 to
1. It is observed that for a given H2' Kr/Fr decreases from 1 at X
= 0, becoming practically constant above a certain X. This trend
can be easily examined, for example, for a given two layered
system (H1, H2 fixed) subject to a wave train of given frequency
(o fixed). At X = 0, the bottom is rigid, hence kr = F means o/k
= (gH1)1/2, which is the well known shallow water dispersion
relationship. Increasing X implies decreasing viscosity v, hence
decreasing p, since r and, therefore, p2 are held constant in
this problem. Note that kr/Fr = C0/C, where Co is the rigid bottom
surface wave celerity and C is celerity at any X>0. Initially,
therefore, as the bottom becomes soft, the wave speed increases
over that due to rigid bottom. As X-, the lower layer becomes
inviscid, and the wave speed equals [g(Hi + H2) 1/2 = [gH1(1 +
2 ]v1/2
H2) ]
Thus, for example, given H2 = 0.2, the ratio Co/C will
approach 0.913 as X increases. For practical purposes, the lower
layer becomes "watery" for values of X exceeding 2 to 3, and the
celerity does not change too rapidly with further increase in X
as seen from Fig. D.1.
In Fig. D.2 the normalized attenuation coefficient, kl/Fr,
is plotted against X for values of H2 ranging from 0.1 to 1. As
before, considering a given system in which only p is allowed to
vary, we note that since kl/Fr = klC0/o, Fig. D.2 essentially









shows how kI changes with increasing X, starting with X=0 at
which k,=0 (rigid bottom case). An interesting feature of the
observed variation in k, is the occurrence of resonance as X
approaches unity. In other words, wave damping is greatest when
the mud layer thickness (H2) equals twice the boundary layer
thickness. Note that the rate of wave energy dissipation is
plgCkla 2 = plgCkia02 exp(-2klx) (Dean and Dalrymple, 1984; Maa,
1986). As X increases beyond this value, ki decreases and
approaches zero as X%- as the lower layer also becomes inviscid.
Characteristic values of X in the present experiment were high.
For example, representative parameters for test 1 are: H2 = 0.28
m, o = 2.51 rad/s and v = 1.76x10-3 m2/s. This gives X = 7.5,
which essentially means that the mud was very soft and resonance
effect was largely absent at the dominant wind wave frequency.
In Fig. D.3, the depth-variation of the velocity amplitude,
um, is plotted corresponding to parameters from test 1, ao = 8 cm
(a somewhat higher than typical value), and X ranging from 0.1 to
1.5 (illustrative range). Notice the heavy damping of oscillation
with decreasing X in the fluid mud layer. By virtue of the model
assumptions, no boundary layer is found in the water layer, even
though the oscillation is damped with decreasing X by virtue of
the momentum coupling of the two layers. Besides the absence of
boundary layer effect particularly just above the interface, the
absence of velocity equality at the interface is yet another
manifestation (limitation) of the assumed inviscid-viscid
behavior. The outcome is suggestion of steeper gradients and
hence rotationality at the interface than in reality, although
laboratory measurements (Maa and Mehta, 1987) do indicate a
rather drastic reduction in the mud velocity relative to that in
water across a thin interfacial layer.
In Fig. D.4, the phase of um relative to the surface wave,
l', is shown for X ranging from 0.1 to 1.5. As with um in Fig.
D.3, phase lags are significant mainly in the mud. A sharp phase
discontinuity is evident at the interface due to the model
assumption which leads to a corresponding discontinuity in the
velocity.









APPENDIX E
MEASURED SPECTRA IN TESTS 1 AND 2


Relevant spectra for test 1 include those for wave energy
(Figs. E.1 and E.2), water velocity (Figs. E.3 and E.4) and mud
acceleration (Figs. E.5 and E.6). These correspond to eight data
blocks (0 hr to 7 hr).
For test 2, measured spectra from data block at 1800 hr are
shown only for illustration. These include wave energy (Fig.
E.7), water velocities (Figs. E.8 and E.9) and mud accelerations
(E.10 and E.11).




















Mobile
Suspenion


-
Lutocline

Tru..ird Mu .
Deforming Bed
Stationary Bed





Fig. 1. Schematic of mud bottom response to waves in terms of
vertical sediment density and velocity profiles (after
Mehta, 1989).





(x,t)



H1 P 2 ,t) Water




z H 2 Fluid Mud


Bed




Fig. 2. Two-layered water-fluid mud system subject to
progressive wave action.















27810'
81000'


2710_'1
80040'1


Test


lo* O


26045
8100'


0 5Km

Depths in Meters
Below Datum


Fig. 3a. Bathymetric map of Lake Okeechobee. Depths are relative
to a datum which is 3.81 m above msl (NGVD).


26045
8040'1
















271 0'
8100,


*0-'O~


26045'
8100o0 Km '-


Mud Thickness in cm


Fig. 3b. Mud thickness contour
Kirby et al., 1989).


map of Lake Okeechobee (after


27010'
80040'


26045'
80040'-
















600



n 500
a-

400

0
1--

3 300
L 3\


r 200

o
r- I
0 100
m



-1.0 -0.5 0.0 0.5 1.0 1.5
WATER LEVEL RELATIVE TO DATUM (m)


Fig. 4. Lake area with mud bottom subject to wave action as a function
of water level relative to datum.







Inner
Shift


Wooden Base
for Data Acquisition
System

Support
Member


ELEVATION VIEW



(







Frame
Members


Accelerometer
Holder


Wooden Base for Data
Acquisition System


N


PLAN VIEW


Base Frame-,


---- 92 cm --
---50 cm


Fig. 5. Tower used in field tests: a) elevation view, b) plan
view.
46


-7-

59 cm








158 cm









50 cm


32cm
32 cm


'-,Pin for
Stability


N -
N -
N
N


A A NN


cm 100 cm


I















































Fig. 6. A view of the field tower,


Fig. 7. Tower and instrumentation assembly
begin deployed at the site.






































Fig. 8, Measurement system in place together with data
acquisition system.


Fig. 9. Bottom core from test 1 is frozen in a mixture of dry
ice and alcohol and cut into 6-8 cm long pieces. Note
the clearly defined mud-water interface.


~


















































1.02 1.04


1.06 1.08


BULK DENSITY, p2(g/cm3)




Fig. 10. Relationship between dynamic viscosity and density for
Okeechobee mud.


103








102








10


1 L
1.00


1.10


1.12








BULK DENSITY (g/cm3)


,1.0


U


-10


-20


-30 -


-40 -



50 "Hard"
-50 -- -


.n
Bottom \


.an I I I


Fig. lla. Mud density profile at the site during test 1.

BULK DENSITY (g/cm3)
1.0 1.1 1.2 1.3 1.4 1.5
0.o I I I
-- ---- s -----.. AC
-0.1 "
Mud
S-0.2 o Layer

< "Hard Bottom"
S-0.3- o
u-
S ----L -- --__
S-0.4"Lens" 0
-0.4 O

-1 0
w -0.5 -

I-o
0. -0.6 -
S ore
o Core I*


Fig. lib. Mud density profiles at the site during test 2.


S Ii



\-


----*--

Mud
Layer










E






z
U
1
C,


w

u



I-
z
C.)






z
CD

s)


TIME (hr)


Fig. 12a. Variation of significant wave height during test 1.


1.0


0.8


0.6


TIME (hr)


Fig. 12b. Variation of modal wave frequency during test 1.


*Surface WVave


Water Velocity






1 I I I I I














(I)
w..
Cu4
II



?
Fn0:
Z w
w >


CC W

LIJ LL
(L CC
(1)


FREQUENCY (Hz)


Fig. 13a. Wave energy spectrum at 1 hr, test 1.


0.2 0.4 0.6 0.8
FREQUENCY (Hz)


Fig. 13b. Water velocity spectrum at 1


250



200



150



100



50


0


I-
LU



SE


-J


UJ
F-
0
a.


hr, test 1.








I 90

z



O
60


,-
Sc

30
0
0



ss


TIME (hr)


Fig. 14. Variation of relative direction
during test 1.

c4
!20

E

> 16 -
0
_J
W 12

U-
LUL



O
0 4-
z


>0 1 2 3 4
TIME (hr)


of water velocity


Fig. 15. Time-variation of water velocity amplitude variance
during test 1.


-
--1--..............










..........................................


I


i


































Fig. 16a.


0 1 2 3 4 5 6 7
TIME (hr)
Time-variations of the variances of horizontal and
vertical mud accelerations during test 1.


1.0

-J
, 00.8
SN

ZE
0*
0.6



SC 0.4
0,


S0.2


Fig. 16b.


0 1 2 3 4 5 6
TIME (hr)
Variations of modal frequencies of horizontal and
vertical mud accelerations during test 1.











0.30


c3
cu 0.24

I-.---
if Z
z 0.18



I-1 0.12-



0) 0.06



0 0.2 0.4 0.6 0.8 1.0
FREQUENCY (Hz)

Fig. 16c. Horizontal mud acceleration spectrum at 1 hr, test 1.


300


cc
L,






Sa-



O-
0
0)


240



180



120



60


0.2 0.4 0.6 0.8
FREQUENCY (Hz)


Fig. 17a. Model calculated
at 1 hr, test 1.


and measured water velocity spectra
























0.30


Oc 0.24 -
N

OE
..0 0.18 Calculated.t I


-- Measured I
wuJ 0.12 -

0uI


0 ,- -
U) 0.06


0 0.2 0.4 0.6 0.8 1.0
FREQUENCY (Hz)




Fig. 17b. Model calculated and measured mud acceleration spectra
at 1 hr, test 1.


i














.. Y
EE
_ U
C3,
ZWUj


0

JL>
a M
Cj,
C),


01 \ I i- I
0 0.2 0.4 0.6 0.8
FREQUENCY (Hz)



Fig. 18a. Wave energy density spectrum at 5 hr, test 1.


300


-J





O

w
0
QO


240



180



120



60


Fig. 18b. Model calculated
5 hr, test 1.


FREQUENCY (Hz)
and measured water velocity spectra at


I
























0.40
-?


Oc< 0.32
N E


0 0.24
I.-II
zW
wu 0.16
SQO

D- 0.08

U3
CL
Cf)


Fig. 18c. Model calculated and
acceleration spectra


0.2 0.4 0.6 0.8
FREQUENCY (Hz)


measured horizontal mud
at 5 hr, test 1.




























LULI
^-- 0.3


O
00-
j.Z 0.2-

Z C Mud Acceleration (Horizontal)

0 0.1- Surface Wave Water Velocity
O


0.00
0 1 2 3 4 5 6 7
TIME (hr)




Fig. 19. Dominant long wave frequency variation during test 1.
























Components of
~ Forcing Wave


f2
-^ kI*-Af


FREQUENCY


Fig. 20. Wave energy spectrum showing short period forcing at
two frequencies and forced long wave.


Forced
Long
Wave

I


- h-Af






















1u



SForcing Wave
5 A AA
1\ I A Forced Wave A \

n,, I II
I I I A ^ I / 1 I
irlr \ r_--^^^^ I .^^^r.--l~- \-^^.-^^^^^^^^-^*^ ---__


- 1 I Ii
II I; /
I V
I V
V


I


V V
v I
v


I I I -


-10L
0


180
Y (deg)


270


360


Fig. 21. Short period forcing wave and forced long wave derived
from water level measurement at 1 hr, test 1.


-. r-


,,










&i *


Fig. B.1 A view of the plexiglass "boat" together with the
accelerometer (not visible). The boat length is 26 cm.
Pen is for length reference only.







*
15




z
UJ
w 10-


-j 5-
0






CQ.


0 5 10 15
MEASURED DISPLACEMENT (cm)


Fig. B.2 Scale measured versus calculated (from acceleration)
wave orbtibal displacements amplitudess) based on
dynamic testing of accelerometer.



















0 0.03 -


c0 0.02 -
U,

I.-


L o p 2 = 1.005 g/cm3
(n 0.01 LH = 0.0026 N.s/m2 -

So!L = 1.014 N-s/m2


0 2 4 6 8 10 12 14

SHEARING RATE (sec1)


Fig. C.1 Relationship between applied stress and rate of
shearing for Okeechobee mud; data for a mud density of
1.005 g/cm3



51 7


4
C., 4
E
z
en 3
LU
C,
w

LU
=T
U,'


p2 = 1.02 g/cm3
H = 0.115 N.s/m2
9L = 1.55 N.s/m2















2 4 6 8 10 12 14


SHEARING RATE (se61)


Fig. C.2 Relationship between applied stress and rate of
shearing for Okeechobee mud; data for a mud density of
1.02 g/cm.


F.












5



4
CV
E
2
cn 3
LUJ

Cn
I--
2

03
CO


0'
0 2



Fig. C.3 Relationship
shearing for
1.04 g/cm.



4



E 3
z

Cn




-0



0 2
o 2


SHEARING RATE (sed1)


between applied stress and rate of
Okeechobee mud; data for a mud density of


8 10 12 14


SHEARING RATE (sed1)



Fig. C.4 Relationship between applied stress and rate of
shearing for Okeechobee mud; data for a mud density of
1.08 g/cm3














4" 4
E o
O
o9 3
(/) o

C2
e-

< P2 =1.10 g/cm3
r iH= 0.211 N.s/m2
1 I tL = 2.90 N.s/m2




0 2 4 6 8 10 12 14

SHEARING RATE (sed1)

Fig. C.5 Relationship between applied stress and rate of
shearing for Okeechobee mud; data for a mud density of
1.1 g/cm3.


LU

I.-


0
O
I-
uL 102






L)
> 1


O


( 1


1.00 1.02 1.04 1.06 1.08 1.10 1.12

BULK DENSITY, p2(g/cm3)
Fig. C.6 Relationship between mud viscosity (relative to water)
and density at "high" and "low" rates of shearing.


1





































X= H2 (Re/2f/2 [=H2(o/2)1/2]


Fig. D.1










II

u"
t-S


Dispersion relationship based on the inviscid-viscid
model.


X= 2 (Re/2)112 [=H2(c/2v)12]


Fig. D.2


Wave attenuation relationship based on the inviscid-
viscid model.


I



































0 0.1 0.2 0.3


VELOCITY AMPLITUDE, urm/s)


Fig. D.3


Simulated profiles of velocity amplitude, ur for
different values of x using parameters from test 1.


PHASE OF urrad)
Fig. D.4 Simulated profiles of the phase of u corresponding to
Fig. D.3


I

































0 Co CO N
ST mC CM
(s-Luo) 3AVM 3aovunlS
:AJISN3Q 7IVE103dS


0o co t r C
wD R cm cm
(s-uwo) 3AVM 3oVdluns
:AJ.SN3Oa IVEl03dS


0








0 -
W



LL



E

C o w I I I 04




o
(s LUO) 3AVM 3ovIunlS

:A0SN3a IlVU103dS



0


N



I ,
z





04JC
0 )

E


0 0 CD Co o
CID r cm C T
(s-z uWo) 3AVM 3ov:lunlS
:AJJSN3a 7Vll103dS


N

I


dO

LUI

U.


N


co;
z
-1--







LLU
cc
U.

































CD co N
(s Luo) 3AVAAM 3oV lnS
:A.USN3=G lVl103dS


0 co tD t 0C
CD T Co C
(s- -o) 3AVM 30VI3lnS
:AJJSN3a IV81033dS


O





co




d -O




N

SE

*0 ti I I
(sC uo) _AVM aoVzu nS
:AIlSN3G 7V il3ocS










N
d0
LL






























o '

E
o o ico Io I
CD T Cr CN














(SE LUo) 3AVM 30V-ilnS
:AIISN3c IVH03SdS
0














U -






0 CO to R! C14
w RT CF CM T-


N
I

C60
z
w
6W
U-
SCC
LL.


N


Cd>"
z
oW
LL



































0 0 0 0 0 0
L 0 tO 0 LO

(s/zwuo) AlOO7-13A E31VM
:AIISN3Ca 7VH.L3dS


0 0 0 0 0 0
IO 0 O 0 0 n

(s/zwo) AIOO13A 831VM
:AJJSN3Q 7VUI103dS


CO
d
N






I
W>



LL


m

4-




a,
o




a)
-4-)

(0


04
al


0 0 0 0 0 0

(s/zwo) AIIoO-13A 83JLVM
:AIISN3Q 1VHO133dS


-4-)
-,









d .
U


N



o w




-
LJ.


O) 8 tO 0 ItO

(s/Lwuo) A31003A U31VM
:ALISN3a -IVOL3dS


I
































0 0 0 0 0 0

(s/LUo) A113lO3A i31.VM
:AIISN3a 7lHllVOUdS


0 0 0 0 0 0

(s/z:uo) AUIIOO13A H31LVM
:AIISN3C "IV.1033dS


CO


zd
o
N




LU

dL
ui
cc
LL,


a 0 a 0 0D0
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Fig. E.11 Vertical acceleration spectrum at 1800 hr, test 2.




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