-
XVIII. INTERACTION BETWEEN SURFACE WAVES AND
MUDDY BOTTOM SEDIMENTS
Joseph N. Suhayda Ports and Waterways Institute Louisiana State
University
Baton Rouge, Louisiana 70803
ABSTRACT
Surface wave-induced bottom pressure fluctuations produce shear
stresses in
soft muddy bottom sediments that cause the sediments to undergo
oscillatory
motion. This motion can be described as a "mud-wave" and causes
surface wave
properties to vary from those that occur over a rigid bottom.
Theoretical studies
have attempted to describe this interaction using a variety of
soil models, i.e.,
viscous fluid, elastic solid, viscoelastic material and
nonlinear viscoelastic.
Although the experimental basis for evaluating the validity of
these assumptions is
incomplete, it appears that a nonlinear viscoelastic soil model
is required to
describe the observed behavior. An example of the interaction of
hurricane waves
and soils found offshore of the Mississippi Delta is considered
in detail. The soil
is described using a model which is nonlinear in relating shear
strain to shear
stress and damping ratio. The surface wave-mud wave interaction
for hurricane waves
is significant and causes wave heights of 70 ft (21.3 m) and 80
ft (24.4 m) in deep
water to decrease to values of from 10 ft (3.0 m) to 25 ft (7.6
m) at a water depth
of 50 ft (15 m). Soil response during this wave-mud interaction
is greatest at
water depths of between 150 ft (45.7 m) and 250 ft (76.2 m).
Maximum soil movements
of 1.5 ft (.46 m) are predicted to occur under hurricane waves.
As a means for
making rough calculations of the wave-mud interaction a
simplified technique for
making engineering predictions is presented. The technique is
based upon a non-
linear stress-strain and damping-strain soil model and predicts
surface wave
attenuation, soil shear stress and shear strain profiles.
INTRODUCTION
This paper presents a review of available information concerning
the inter-
action of surface waves and muddy bottom sediments. Muddy bottom
sediments occur in
a variety of coastal zones in the United States and in the world
such as the
Guianas, the northern coast of China and in southwest India. The
presence of muddy
bottom sediments has a profound effect on hydrodynamic
processes, particularly on
surface waves. The purpose of this paper is to document the
current state of
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knowledge of wave-mud Interactions, demonstrate the significance
of this interaction
in offshore design wave forecasts and to present a simplified
technique for pre-
dicting the attenuation of surface waves propagating over muddy
bottom sediments.
MUDDY COASTS
Muddy coasts represent a type of coastline where fine grained
sediment is
consistently present in the nearshore waters and on the
shoreline. This type of
coast occurs at all latitudes and is often associated with large
deltas, lagoons and
estuaries. In general, the offshore area has a smooth, low
sloping profile, with
very turbid water occurring from the shoreline to several
kilometers offshore.
Usually a mud flat, which is exposed at low tide, occurs in
front of a vegetated low
backshore. In the United States, muddy coasts occur extensively
in Louisiana and
Florida.
Morphology
Coasts having extremely high concentrations (i.e., 10,000 mg/1)
of suspended
material in the nearshore water can have a wide range of
geometries and forms. Some
of the most common shoreline features are the presence of
nearshore mudbanks, beach
ridges and vegetation, i.e., tidal marsh, meadow, mangroves or
nipa palms. The
vegetation is usually fronted by an unvegetated flat. There is
rarely any coarse
material on the flats or forming beaches along the shoreline. A
small scarp may
occur at the shoreline if the coast is undergoing erosion and
retreat. Coarse sedi-
ment may occur as a deposit at the base of the scarp. When
storm, hurricane or
typhoon waves attack the coast, this coarse lag is cast over the
vegetated marsh and
forms thin linear detrital deposits. In addition to the coarse
lag, erosion of the
vegetated shoreline yields large quantities of organic debris
that adds to the
material in suspension in offshore waters.
A second kind of muddy coast consists of bare mud flats with
only a few
scattered salt tolerant plants on its surface. This shoreline is
most commonly
found in high tide regions or in areas where severe climates
prevail (arid, semi-
arid or arctic). The shoreline position constantly changes as a
result of water
level variations induced by tides, wind setup or atmospheric
pressure changes.
Slopes on these flats are extremely low, ranging from 0.01 to
0.0005. A variety of
drainage patterns can be found on the flats, whose configuration
changes due to the
influence of tides and coastal currents. On some bare mud flats
thin narrow beach
ridges composed of sand or shell debris are found behind the
normal high tide
shoreline.
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Offshore of muddy coasts the bottom may be covered by patches of
"fluid mud".
This "fluid mud" is a thixotropic gel formed when suspended
sediment concentrations
become very high, between about 10,000 and 250,000 mg/1 (40).
This mud has the
consistency of yogurt and remains as an intact mass for long
periods of time (i.e.,
months to years). Where fluid mud is present the bottom may be
difficult to define
because mud concentration and soil density gradually increases
from the water column
to the underlying consolidated sediment.
The concentration of fine grained sediment in the water column
of a muddy coast
can undergo exceptionally large variations in short periods of
time. Sediment con-
centrations in nearshore waters of muddy coasts are given in
Table 1 (40). In the
Gulf of Thailand, suspended sediment concentrations at middepth
in 10 m of water
varied from 52,000 mg/1 during ebb tide, to 2,100 mg/1 during
slack tide, to 6,000
mg/1 during flood tide. Waves have caused suspended sediment
concentrations to vary
from 1,500 mg/1 during low waves (wave height 1 m) to 6,200 mg/1
during high waves
(wave height 4 m). These high concentrations of sediments can
cause a significant
increase in the viscosity of the water.
Wave-Bottom Interactions
As surface waves propagate over muddy bottom sediments an
interaction occurs
which does not occur when waves pass over a sand or rock bottom.
This interaction
involves the physical movement of the muddy sediments in a mass.
Waves cause bottom
pressures which may be larger than muds can support. Under wave
action the bottom
muds are alternately exposed to high and low pressures, that
cause the muds to
oscillate. This movement can be visualized as a "mud-wave", as
illustrated in
Figure 1.
The height of the mud-wave depends upon the geotechnical
properties of the muds
and the amplitude and wavelength of the bottom pressures.
Heights of mud-waves
range from a fraction of a centimeter under low surface waves to
a meter under storm
waves. The mud-wave can have a significant effect on the surface
wave because it
represents a boundary which moves and can absorb energy. Very
large losses of
energy from surface waves can occur when mud-waves are
generated. The wave height
can decrease by 10% in a distance of as little as a few tens of
meters.
REVIEW OF THE LITERATURE ON WAVE-MUD BOTTOM INTERACTIONS
The present understanding of wave-mud interaction has developed
over the last
30 years. The first studies occurred during the 1950's and were
based upon field
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TABLE 1. ( a , b)
a. SURFACE SUSPENDED SEDIMENT CONCENTRATION
IN NEARSHORE MUDDY COASTAL WATERS
Location Concentration (mg/1)
Maximum Minimum
Louisiana Coast 6.4 x 101
1.0 x 10
East China Sea 7.0 x 101
5.0 x 10
Venezuela Coast 1.0 x 102
1.0 x 10
Gulf of San Miguel 2.0 x 102
6.0 x 101
Dutch Wadden Sea 6.2 x 102
5.0 x 101
Gulf of Thailand 9.7 x 102
1.0 x 10
Gulf of Ho Pai 1.0 x 103
1.0 x 102
British Guiana Coast 2.6 x 103
5.0 x 10
3 1 Surinam Coast 3.7 x 10 4.5 x 10
b . SURFACE SUSPENDED SEDIMENT CONCENTRATIONS IN RIVER
AND ESTUARINE WATERS ALONG MUDDY COASTS
Location Concentration (mg/1)
Maximum Minimum
Harlingvliet Estuary 1.2 x 102
3.8 x 101
(Netherlands)
Po River Plume 1.1 x 102
7.0 x 10
Ems Estuary 1.8 x 102
1.0 x 101
Thames Estuary 2.0 x 102
1.0 x 10
Mississippi River 3.1 x 102
4.0 x 10
(South Pass)
Chao Phya River 6.9 x 102
1.4 x 101
Surinam River 9.2 x 102
6.0 x 10
Bristol Channel 1.3 x 103
3.0 x 101
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Figure 1. Schematic diagram showing the surface wave and the
"mud-wave" along the water sediment interface.
observations by earth scientists. The interest in wave-mud
interaction signifi-
cantly increased as a result of the development of offshore
sites for oil
production. Mud-wave interactions were and are an important
consideration in
developing design criteria for offshore oil production
platforms. Theoretical and
laboratory studies were conducted and some predictive models
were developed during
the 1970's. Present research is focused upon conducting field
and laboratory
experiments to test existing predictive models and upon
developing new predictive
models.
The first reference to wave interaction with soft bottom
sediments is Ewing and
Press (14). It was reported that a kind of wave motion was
likely to occur in soft
sediments and that it would affect the characteristics of the
overlying surface
waves. They suggested that a "liquid" bottom could change the
magnitude of wave-
induced bottom pressures from the values which would occur over
a rigid bottom.
Gade (17), theoretically investigated wave-induced bottom
sediment movement
with the sediment modeled as a viscous fluid. Small amplitude
waves were assumed to
exist at both the sea surface and at the water/sediment
interface or mudline. The
model predicted that the surface wave would decay exponentially
with distance
traveled. The rate of decay had a maximum value when the
dimensionless ratio,
1/2
h/(a/2v) , had a value of 1.2, where a is the angular frequency
of the wave, v the
sediment viscosity and h the thickness of the mud. Examples were
presented com-
paring wave decay over a sand bottom with wave decay over a mud
bottom. For a wave
with a height of .61 m and a period of 8 sec in a water depth of
1.22 m , the wave
height would be reduced to 79% of its original value after
traveling 300 m over a
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sandy bottom. Over a mud bottom only .46 m thick and having a
viscosity of .46
m /sec, the same wave would be reduced to less than 37% of its
original height after
traveling the 300 m . For a thickness of mud of 1.83 m the wave
height would be
reduced to 2% of its original height after traveling 300 m . It
was concluded that
even for a relatively thin layer of fluid-like mud the
dissipation of wave energy to
bottom motion viscosity effects exceeds by far the energy loss
due to bottom
friction. Effect due to muds having some rigidity were addressed
in later studies.
Subsequently the problem was investigated with sediment modeled
as a linear
viscoelastic material or a Voigt solid (Gade (18)). The material
properties which
specify a Voigt solid are an elastic modulus and a viscosity. It
was found that the
amplitude of the mudline wave was determined by the elastic
properties of the sedi-
ment and was not noticeably influenced by sediment viscosity.
However, it was also
found that sediment viscosity determined the rate of dissipation
of wave energy.
The values used for sediment shear modulus and viscosity, the
Voigt constants, were
hypothetical since no actual data were available.
Henkel (20), examined the conditions under which surface waves
could cause soft
sediments to fail at large depths (tens of meters) below the
mudline. Using linear
wave theory over a rigid bottom to estimate wave-induced bottom
pressures he found
that hurricane storm waves could cause Mississippi Delta
sediments to fail at depths
as great as 30 m . Sediment shear strengths in the delta area
were described as
increasing at an approximately linear rate of 2-4 psf per foot
(1.5 to 3 kPa per m)
of depth below the mudline.
Mitchell (28) conducted laboratory studies of wave induced
sediment insta-
bility. The wave induced shear stresses when combined with
gravity stresses
produced mass movement of sediment over about two thirds of the
soil profile.
Because of the failure of an oil platform off of the Mississippi
Delta during
hurricane Camille, 1969, due to a submarine mudslide, the
interest in wave-mud
interactions involving extreme wave heights (2,3) significantly
increased during the
1970's. The immediate need was to predict the amount of mud
motion occurring under
hurricane waves. Several reports are available which documented
the affect of wave-
induced mudslides on offshore structures
(1,4,6,7,23,32,34,43).
Wright and Dunham (42) presented a simplified finite element
model to study the
wave-induced soil motion problem. The model Is static,
two-dimensional and includes
effects of large vertical deformations of the seafloor boundary
as well as the non-
linear stress-strain characteristics of the sediments. Input to
the model are the
bulk modulus and its variation with increasing numbers of stress
cycles and stress
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magnitude. Sediment vertical motion of .76 m was predicted to
occur during hurri-
cane wave conditions at the seafloor in a water depth of 100 m.
Even at a depth of
30 m below the seafloor the predicted vertical sediment motion
was about .3 m.
Laboratory tests were reported by Doyle (13) on wave interaction
with soft
sediments. It was reported that bottom sediment movement showed
a considerable
variation with depth below the mudline. Wave-induced bottom
pressures were consid-
erably altered from those expected over a rigid bottom. Measured
bottom pressures
were only about 27% of predicted values. It was reported that
considerable remold-
ing of the sediments took place in the zone of soil movement.
The distribution of
wave-induced shear stress with depth was given indicating that
the maximum stress
was about .37 times the bottom pressure and occurred at a depth
of .16 times the
wavelength.
Carpenter, Thompson and Bryant (8) described the viscoelastic
properties of
marine sediments. In this study the shear resistance of the soil
was related to the
rate of shear strain. The soils were shown to behave as
non-Newtonian viscous
material. The viscous property was indicated to be related to
the liquid limit of
the soil at a depth below the mudline.
Esrig, Ladd and Bea (15) presented the results of a laboratory
study of
Mississippi Delta sediments under simulated wave loading. Data
relate the shear
strength of the sediments and their elastic properties and the
liquidity index.
These data indicate that the shear strength decreases as a
function of decreasing
liquidity index. The ratio of the elastic modulus and the shear
strength is shown
to be a nonlinear function of axial strain and the number of
cycles of wave loading.
The ratio decreases with increasing number of cycles and
increased strain.
The development of a sediment failure profile was investigated
by Pamukcu and
Suhayda (31) using a numerical wave/sediment model. The soils
where modeled using
critical state parameters so that stresses, strains and soil
property changes could
be described for the first few cycles of loading. It was found
that soil properties
under waves could be forced to the critical state after from one
to a few cycles of
wave action.
The first reported field measurements of wave-induced seafloor
motions were
presented by Suhayda, et al. (36). Simultaneous measurements of
bottom oscillations
and wave characteristics were made in East Bay, Louisiana, in a
water depth of
19.5 m. Bottom motions were small (< 1 cm) under waves having
a height of 1 m and a
period of 5 sec. The bottom sediments moved in an elastic
response with the
seafloor being depressed under the wave crest, that is, the mud
acceleration was
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upward under the wave crest. Additional analysis of this data
was presented in
(41).
The first reported field measurements of wave decay over muddy
sediments were
presented by Tubman and Suhayda (38). The wave induced bottom
pressure, amplitude
and phase of the mudline wave and the decay of the surface wave
were measured in
East Bay. The energy loss rate was about 100 times that
predicted for a sandy
bottom. The actual work done on the mudline wave was measured
and found to agree
with the magnitude required to explain the anomalously high wave
decay. The bottom
pressure wave was found to be out of phase with the bottom mud
wave by about 202
degrees, so that the trough of the mud-wave was about 22 degrees
out of phase with
the pressure wave crest.
Wells (40) presented the results of field studies on the
extensive mud flats of
Surinam. It was found that wave heights in very shallow water
maintained a height
to water depth ratio, H/h, of .23, due to the effect of wave
attenuation by the soft 2
mud. A mud viscosity of up to 210 cm cm/sec (20,000 times the
viscosity of water)
was measured for the fluid muds. The water content of the mud
was about 380% (ratio
of the weight of water to weight of solids) and had a specific
gravity of 1.17. The
distance over which the wave height decayed by 10% was between
150 and about 600 m.
Several theoretical studies have addressed wave propagation over
a deformable
bottom. These studies will form the basis for the methodology to
be developed in
this study. Mallard and Dalrymple (26) developed an analytical
solution in which
the soil beneath the water was regarded as an elastic solid.
They showed that the
phase speed of the surface waves would be modified from that
given by classical
rigid bottom theory.
Suhayda (37) derived the linear wave theory over a bottom
boundary that is not
rigid. The bottom movement was assumed to be described by a wave
of the same
frequency as the surface wave but having an arbitrary phase
shift. The results show
that mudline waves can cause changes to wave properties such as
the velocity profile
and dispersion over a moving bottom boundary.
In an additional paper, Dalrymple and Liu (10) treated the
bottom muds as a
viscous fluid, as had been assumed by Gade (17). The problem was
solved for inter-
mediate as well as shallow water. Comparisons with the
laboratory experiments of
Gade (17) showed that a mud viscosity could be found which would
produce agreement
between the observations and the predictions of the theory. The
kinematic viscosity 2
of the mud required to do this was .0026 m m/s or about 1,000
times the viscosity of
water.
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The effect of soil inertia was investigated by Dawson (11),
where it was shown
that for an incompressible sediment, the wave speed, particle
velocity and pressure
were affected by the soil inertia.
Hsaio and Shemdin (22) considered the muds as a viscous fluid
and compared wave
attenuation caused by muds with the attenuation occurring on
sandy coasts, i.e.
bottom friction and percolation. The wave attenuation rates
caused by muds were up
to 100 times larger than either bottom friction or percolation.
Solutions were not
developed into useful forms, but were presented graphically.
Schapery and Dunlap (32,33) presented a method for predicting
wave induced
seafloor motions using a viscoelastic model for bottom
sediments. This method has
been utilized extensively by the oil industry to make
predictions of seafloor
motions during hurricanes. The important soil parameters
required are the shear
strength of the soil, unit weight, liquidity index, shear
modulus at small strains,
nonlinear stress-strain behavior and hysteretic damping of the
soil. The method has
been incorporated into a computer model and predictions of wave
attenuation as a
function of distance can be made on a site specific basis. This
computer model is
proprietary and was not intended to be used to make shallow
water wave height
forecasts during hurricanes. It does not include the effects of
wind Input of
energy, for example.
The elastic properties of submarine sediments have been the
focus of recent
work by Coleman, Dawson and Suhayda (9) and Dawson, Suhayda and
Coleman (12). They
used previously measured sediment movement data to determine the
insitu shear
modulus of marine sediments in the Mississippi Delta. Using wave
theory for waves
on an elastic bottom, the calculated shear modulus was 3,000 to
6,000 kPa. The
shear modulus decreased with increasing wave frequency. The
ratio of the shear
modulus to the shear strength of the sediments under non-storm
conditions was about
100.
Forristall, et al., (16) reported the first measurements of
storm wave decay
for the Mississippi Delta. Measurements were made of wave
heights during hurricane
Frederic at a station in 312 m of water and at a station in 20 m
of water in East
Bay, Louisiana. The deep water significant wave heights reached
a peak of about
8.4 m while at the same time the waves in East Bay reached only
1.7 m. Refraction
analysis was reported to change the directional spectrum by
about 20 percent. Wave
shoaling, for shallow waves, increased wave height. These
results clearly
demonstrated that wave decay previously observed during moderate
conditions also
occurs during hurricane storms.
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A recent theoretical paper by MacPherson (25) has considered
wave attenuation
over a seafloor consisting of a linear viscoelastic material.
Water depth and
sediment thickness can take on any value, rather than be limited
to shallow water.
Several different types of wave-bottom interaction were
identified, depending upon
the relative importance of the sediment elastic and viscous
properties. The paper
presents the process of wave attenuation as an exponential decay
of wave height
given as a function of input wave parameters, water depth and
soil properties.
Yamamoto (44) has described the interaction of a spectrum of
waves with bottom
sediments having poro-viscoelastic properties.
Pamukcu (29,30) presented profiles of sediment dynamic
properties of material
from a boring taken in the Mississippi Delta. The shear modulus
was found to depend
upon the magnitude of shear strain, so that the sediment becomes
less stiff as it is
moved. The shear modulus was found to be about 250 times the
shear strength of the
sediment.
Kraft, et al. (24) presented the results of predictions of soil
response to
ocean waves. The predictions were made using the model developed
by Schapery and
Dunlap (32) and soil properties typical of the Mississippi
Delta. Results of the
study indicate that the wave induced stresses on the muds can
alter the soil prop-
erties during the loading event. Thus soil properties at the
initial stage of a
storm may not be the same as during the later stages of the
storm. Quite large soil
displacements are predicted during extreme wave conditions.
Summarizing the state of knowledge of wave-mud interactions, it
is evident that
while there is a great deal of isolated theoretical and
experimental work completed,
there is not available at this time a comprehensive theoretical
description of the
interaction which has to be verified with conclusive experiments
in the laboratory
and in the field. The material properties of the muds that are
critical for accu-
rately describing the interaction are both viscous and elastic
and depend in a
nonlinear way upon axial and shear strains, axial and shear
rates of strain and
number of cycles of loading. The theoretical models have not
included nonlinear
material properties in the analysis. The theoretical models
indicate rates of wave
decay can be very large for certain assumed values of input soil
properties.
However, these properties have not been well documented for many
field sites. What
little documentation of these properties that exists has been
primarily in the area
of the Mississippi Delta. For this area there seems to be a
correlation between the
viscoelastic properties of the sediments and standard soil
parameters such as shear
strength and liquid limit. This information is used in the model
presented in this
paper.
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DATA ON WAVE DECAY OVER MUDS
There have been relatively few measurements made of wave decay
over muds,
although the range in wave heights involved is considerable.
There have been a number of reports of hurricane induced wave
heights in the
shallow water areas of Louisiana. Unfortunately, most of the
data are the result of
visual estimates or inferred wave heights. The wave data which
follow represents a
survey of available wave data roughly in the order of decreasing
importance.
Bea (3) and Bea and Aurora (5) reported wave heights occurring
within East Bay
Louisiana during hurricanes Betsy, 1965 and Camille, 1969. The
wave heights were
estimated by experienced engineers from the damage sustained on
offshore platforms.
These hurricanes are considered great hurricanes having
respective central pressures
of 27.8 and 26.7 inches of Hg. The path of both of these storm
was northward
directly toward East Bay. Betsy had a landfall to the west of
East Bay, while
Camille veered eastward having a landfall along the Mississippi
coast near Pass
Christian. The storms each passed within 20 miles of East
Bay.
Bea (3) reported maximum wave heights in East Bay resulting from
hurricane
Camille at several locations. At a water depth of 100 m, the
maximum wave height
was 20 m. Near South Pass of the Mississippi river, where the
water depth was 12 m,
the observed maximum wave heights were less than 4.5 m. For a
wave height equal to
4.5 m, the wave height to water depth ratio would be 4.5/12 or
.375. Within East
Bay, where the water depth was also about 20 m, the maximum wave
height was less
than 3 m. The wave height to water depth ratio was at most
.167.
Within East Bay during hurricane Betsy, the maximum wave height
was estimated
to be about 7 m at a water depth of 20 m, giving a wave height
to water depth ratio
of about .35. The offshore maximum wave height during hurricane
Betsy was measured
to be about 17 m, at a water depth of 76 m. The deep water
maximum wave height for
Betsy was estimated to be about 23 m.
These results are very important for several reasons. These
storms were of
extreme intensity and passed very close to an area of offshore
Louisiana where a
considerable amount of information is available concerning
bottom sediment geologic
and geotechnical properties. Also, offshore wave height
measurements were made at
several locations near East Bay and estimates of maximum wave
heights were reported
in shallow water. These data sets can be used to test the
validity of any theoreti-
cal models developed to predict the decay of surface waves over
muds.
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Additional measurements of hurricane wave heights in East Bay
have been
reported as part of the SWAMP study (16) (Forristall, et al.,
1980). During this
project wave heights were measured over several years at the
Cognac platform and at
various locations in East Bay. The Cognac platform is located
about 27 km southeast
of East Bay at a water depth of 312 m. The most Important data
collected during the
study was from hurricane Frederic, 1979. Hurricane Frederic had
a central pressure
of 27.8 inches of Hg and passed within 150 km to the east of the
platform. The
maximum significant wave height measured at the platform was 8.4
m while the maximum
significant wave height measured at a water depth of 33. 5 m,
was 1.7 m. Wave
spectra were presented from the deep water and the shallow water
measurement sites.
These data are for a hurricane that is as intense as the 100
year storm, however it
passed to the east of East Bay. These data do provide direct
measurements document-
ing wave height decay in the area of East Bay.
Additional shallow water wave data is available for areas of
Louisiana to the
west of East Bay. These data are not well documented, but they
do give some
indication of the observed extreme wave heights. The best of
these data are from
Bay Marchand, Louisiana, having a normal water depth of 2.7 m
and a storm water
depth of 5.9 m. The maximum wave crest elevation observed was
4.7 m above mean Gulf
water level. This corresponds to a maximum wave height of about
2.1 m or a maximum
wave height to water depth ratio of .37. The wave crest to wave
height ratio for
this wave was taken to be .7.
Other wave data have been reported for various locations in
Timbalier Bay,
Louisiana, resulting from hurricane Hilda, 1964. The data
represent visual observa-
tions made at different oil fields within the bay by workers
during hurricane
conditions and reported to their various companies. Hurricane
Hilda had a central
pressure of 28.3 inches of Hg with deep water maximum wave
heights of 16-17 m. The
hurricane had a landfall near Saint Mary parish. The
observations included wave
height and storm water level and are presented in Table 2. The
table gives the
location of each observation site, the wave crest elevation and
the mean water
elevation above the mean Gulf water level. Also given is the
computed maximum value
of the ratio of the wave crest elevation above the storm water
level to the storm
water height. Normal water depths at the sites are not readily
available, so that
the actual water depth during the storm are not known. The data
therefore provide
estimates of the maximum wave height to water depth ratios. The
data indicate that
the maximum wave height to water depth ratios reached an average
value of about 1.0
for locations near the normal shoreline and decreased to a value
of less than .3 for
locations at the interior shorelines of coastal bays.
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413
TABLE 2
WAVE DATA AT VARIOUS LOCATIONS FOR HURRICANE HILDA, 1964
Location
Water Elevation
(ft)
Wave Crest Elevation
(ft)
Min. Wave Height Ratio
Bay Junop Field 6 9 .71
Bay St. Elaine Field 8 10 .36
Caillou Island 4 8 1.43
Dog Lake Field 6 9 .71
Golden Meadow 6 8 .47
Lake Barre Field 8-10 10-13 .47
Lake Pelto 8 10 .36
Lake Raccourci 8-10 10-13 .47
Ladeyrouse Field 5 7 .57
Leeville Field 5.5 7 .43
Montegut Shipyard 5 7 .57
Note: Ratio is wave crest the water elevation.
elevation minus water elevation divided by .7 times
DEVELOPMENT OF A SIMPLIFIED PREDICTION MODEL
Several models exist for predicting the response of soils to
surface waves and
involve a variety of assumptions regarding wave and soil
properties. The model
presented here is intended as a tool for making engineering
forecasts of wave-mud
interactions. It is simplified from the model of Schapery and
Dunlap (91) and is
intended to give reasonable estimates of the primary features of
the wave/mud inter-
action, i.e., wave attenuation and mud stresses and strains. It
is based upon a
description of the underlying physical processes that include
several simplifying
assumptions (5).
Basic Hydrodynamlc Equation
The model is based upon the conservation of energy for waves
given by
d(E Vg)/dx = Q (23)
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414
where E is the wave energy density in joules/m, Vg is the group
velocity in m/s, x
is the coordinate positive in the direction of wave propagation
in m and Q repre-
sents the rate of energy loss or gain from processes in joules/m
s. An expression
for the conservation of wave energy in terms of wave height can
be written given
that
E = pgH2
/8 (2)
and
d(H2
Vg)/dx = 4Q/pg (3)
The process of wave decay due to wave-mud interaction represents
loss of wave
energy. This loss has been modeled as an exponential decay by
MacPherson (25). For
this process the rate of energy exchange is proportional to the
wave energy present,
that is
4Q/pg = A H2
(4)
where A is a proportionality constant. The solutions of the
energy equation for
these processes show exponential growth or decay with distance,
that is
H(x) - H(0) EXP(A x/2Vg) (5)
and for
Ka = EXP(A x/2Vg) (6)
and
H(x) = H(0) Ka (7)
Mud Bottom Attenuation
The effects of muds on wave propagation are computed from a
formula based upon
the results presented by MacPherson (25). What needs to be
determined is the atten-
uation parameter A in equation 5. The attenuation parameter is
calculated from a
formula developed for various bottom sediments of the
Mississippi Delta. The
attenuation parameter for mud is (37):
A = -3.14 MS SIN(
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415
speed. The bottom movement parameter MS is the ratio of the
amplitude of the mud-
line wave to the amplitude of the surface wave. The formula used
from MacPherson is
equation 3.33 on page 729. It depends upon the phase speed of
the wave and the
dynamic properties of the shelf sediments. Values of MS are
computed from a formula
which was derived from equation 3.33. The derivation is as
follows:
1/2 2(gh) |(PjgG + i P L P 2 g v o ) |
** r
2. 2. 2 ^ 2 . 2 2. (9)
O ( P24 O (V + G / p
2a )J
Then using y - p2v and C = ( g h )
1
/2
2 C | ( pl gG + ipjgy)|
MS ( 1 0 ) 4 a (y V + G )
MS Cpjg|(G + ipa)|
j 2 2 2a(G + y a )
(11)
MS C P
xg
2a(G2
+ 2 2.1/2
y a ) (12)
32 C T M S
" , ,r2 ^ 2 2.1/2 ' (13)
2ir(G + y a )
The formula for MS, with C T = L , where L is the wavelength in
m is
MS - 5.09 L/GE (!4)
where
GE - (G2
+ y V )1 / 2
( 1 5 )
The phase angle between the mudline wave and the surface wave is
computed from
equation 3.34 in MacPherson,
TAN() = ya/G (1 6 )
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416
The phase angle Is related to the damping ratio Dt given by
Dt = TAN(/2) (17)
From the work on the stress/strain properties and energy loss
from muds and clays,
(29), the damping ratio and the shear modulus of marine clays
have found to be a
function of the shear strain. The formulas relating the shear
modulus and the
damping ratio to the shear strain are:
G = GS = GM/(1 + RA y) (18)
and
Dt = FI0(1 + RA Y) (19)
F I O = A r c T a n ( . 1 2 LD (l + (GM Y /SU)) ( 2 0 )
The shear modulus and viscosity are computed from nonlinear
formulas as a function
of shear strain in the soil. The formulas as a function of shear
strain are
GS = GM/(l + (GM Y/SU)) (21)
y = (GS T/6.28) (.12 LD 1 + (GM Y/SU)) (22)
GM = RA SU (23)
where GM is the initial tangent modulus, SU is the undrained
shear strength, Y Is
the shear strain in the soil, LD is the liquidity index and RA
is the ratio of
GM/SU. The liquidity index is related to the shear strength by
(15)
LD = l.l/(.8 + SU/500) (24)
The shear strain in the soil is computed from the following
formula
Y = Tw/GE (25)
where Tw is the wave induced shear stress in the soil. This
shear stress in the
sediment depends upon the wavelength of the wave and the bottom
pressure amplitude
(13).
Tw = PA (6.28 z/L) e -(6.28 z/L)
( 2 6 )
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417
where z is the distance (positive) below the mudline. The
maximum value of the
shear stress for a constant shear modulus profile is .37 Fa and
occurs at a depth of
.16 L.
If the shear strength varies linearly with depth, given by
SU = SO + Z(S1 - S0)/D1 (27)
then the maximum shear strain occurs at a depth ZD given by
Z D - T l s m n r T ) H 1 - H r ( S i / s o - i ) } 1 / 2 - i ]
( 2 8 )
where SO and SI are the shear strengths at the mudline and at
depth of D1 below the
mudline and L is the wavelength.
Measurements of the value of RA are rare and vary over a large
range. Values
from laboratory measurements range between about 30 and 700.
Field measurements
under storm conditions are not available, however measurements
made during the
SEASWAB experiment, as presented in (22), indicate a value of GM
in the range of
3,000 to 6,000 kPa for a site in East Bay. For the shear
strengths found in East
Bay, this implies a value for RA of 100.
From a given value of the shear strength at the mudline and at
some depth below
the mudline, the attenuation coefficient for wave decay over
muds can be calculated.
Wave induced bottom pressures at a site are calculated using
linear wave theory.
The bottom pressure amplitude is
Pa - H w/(2 Cosh(kh)) = H w CP/2 (29)
where H is the wave height and w is the unit weight of sea
water, k is the wave
number, CP is the pressure coefficient and h is the water depth.
The wave length Is
calculated from linear wave theory.
A forecast for East Bay, Louisiana is presented in Table 3. The
significant
wave heights in East Bay are limited to maximum values of less
than 10 ft for
nominal water depths less than about 50 ft. If the same forecast
is made with the
mud attenuation removed the significant wave heights are about
twice as large. It
has been found that the effects of muds become very significant
for values of the
shear strengths below 200 psf, they show moderate effects at 400
psf and little
effect above 800 psf.
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418
TABLE 3
COMPARISON OF PREDICTED AND OBSERVED WAVE HEIGHTS IN EAST
BAY
Observed Predicted
Water Depth
(ft) Hm (ft)
Wave Hgt. Ratio
Hm Wave Hgt (ft) Ratio
60-70 20-25 .35 22.4 .34
60-70
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419
linear theory for these sediment characteristics, as shown in
Figure 8. The ratio
of the undrained shear strength to the average maximum shear
stress along the wave-
length can be used as a safety factor against large seafloor
motions. The variation
of this safety factor along the transect is shown in Figure 9.
The safety factor is
close to one for water depths between 100 and 250 ft (30 and 75
m).
The wave-induced soil motions are described in Figures 10-12.
The horizontal
soil movements under hurricane waves are predicted to be up to
1.5 ft (.5 m ) in
magnitude, depending upon the shear strength profile used as
shown in Figure 10 and
11. Shown in Figure 12 is the profile of the horizontal soil
movements. Maximum
soil movement occurs at depth below the mudline of about 20-30
ft. This is a
shallower depth than expected for the case of a constant shear
strength half space.
Motion is predicted to occur to a depth of over 150 ft (30
m).
The relationship between the surface wave profile and the bottom
pressure and
mud wave is shown in Figure 13-15. The bottom pressure is
shifted from being in
phase with the surface wave, as it would be over a rigid bottom,
to a condition of
lagging the wave crest by about 90 degrees when the ratio of the
shear modulus to
the shear strength decreases to a value of 50, as shown in
Figure 13. The hori-
zontal and vertical mud motions also show a large phase shift
and an amplitude
increase for the same G/Su value of 50, as shown in Figures 14
and 15.
UNDRAINED SHEAR STRENGTH, su ,KSF 0 0.4 0.8 1.2 1.6 Offci i i |
i i i | i ii | i i i
0
u.
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420
LIQUIDITY INDEX
Figure 4. Generalized profile of the liquidity index for various
water depths.
Reproduced from (24) by permission of Crane, Russak &
Company, Inc.
Figure 5. Data showing the nonlinear relationship between the
soil shear stress and
damping ratio as a function of strain. Reproduced from (24) by
permission
of Crane, Russak & Company, Inc.
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421
120
100
5 60 UJ I
UJ I 4 0
20
J 1 1 1 1 1 1 1 1 1 1 1 1 1 | II 11_ STRENGTH PROFILE VARIED
WITH. WATER D E P T H (CASE 1)
STRENGTH PROFILE FOR 3 0 0 F T m mm WATER DEPTH USED FOR A L
L
_ WATER D E P T H S ( C A S E 2 ) -
_ s f -- (jL^- T=8sec = 1 / 1 '7 / / V
_
/ / / SL -
/ / J / 14 sec -/ 4 /
~l 11 i 1 M 1 1 1 1 1 1 1 1 1 i F 100 200 300 4 0 0 W A T E R
DEPTH,FEET
Figure 6. The computed decrease in storm wave height resulting
from wave/mud interaction. Reproduced from (24) by permission of
Crane, Russak & Company, Inc.
>1 I i I I I I I I UJ o
t 8 0 0 1 _i CL 2 < 600 UJ a.
400h UJ cr CL
200
I I
I I I I | I I I I _
W A V E W A V E HEIGHT.ft PERIOD.sec -
70 14 60 16
I I l I ' I l ' ' ' I ' ' ' I
100 200 300 W A T E R DEPTH, FEET
400
Figure 7. The variation with water depth of the bottom pressure
amplitude for storm waves. Reproduced from (24) by permission of
Crane, Russak & Company, Inc.
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422
Figure 8. Comparison of the wavelength as a function of wave
period for various water depths according to linear wave theory
(lines) and mud bottom theory (symbols). Reproduced from (24) by
permission of Crane, Russak & Company, Inc.
Figure 9. Safety factor as a function of water depth for storm
waves. The safety
factor is the ratio of the average bottom pressure amplitude to
the shear strength of the bottom sediments. Reproduced from (24) by
permission of Crane, Russak & Company, Inc.
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423
| 2.0
s 1.5
: i i i i I i
i.o
<
8 Q 5
t-
r | i i i i | I I I I _
T=l6s)H
0=60FT-
T= I4s,Ho=70FH T=l2s,H
0=70FT-
T= I0S,Hq=80FT~
I I I I 100 200 3 0 0
W A T E R D E P T H , FEET 4 0 0
Figure 10. Magnitude of the predicted horizontal movement of
bottom muds under storm waves. Reproduced from (24) by permission
of Crane, Russak & Company, Inc.
Figure 11. Comparison of the magnitude of mudline soil movements
between the depth variable shear strength profiles shown in Figure
2 (case 1) and a shear strength profile for all water depths equal
to the 300 ft profile in Figure 2 (case 2). Reproduced from (24) by
permission of Crane, Russak & Company, Inc.
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424
MAXIMUM HORIZONTAL SOIL MOVEMENT, FEET
50
100
o: O o u. < UJ
? 150 S UJ m
200 a. UJ a
250
300
T T T T T T T n ]TTTT[ I i n r j m - y -- 7 FT - ^ J :
- / >WAVE HEIGHT I - / r / / 18.5 FT - / / -- / / -: / / /
/
M '/
- / SOIL CONDITIONS I
CASE 1 I
_ CASE 2 ~
-50 FT WATER DEPTH
-14 SEC PERIOD ~
m i l Lit 111 ii I n n , h i i f
Figure 12. Shows the profile of horizontal soil movements at a
site in 50 ft of water for case 1 and case 2. The stronger offshore
soil strength profile (case 2) shows larger movements at the site
because the wave height is larger. Reproduced from (24) by
permission of Crane, Russak & Company, Inc.
77" 277
PHASE ANGLE,RADIANS
Figure 13. Shows the magnitude and phase relationship between
the surface wave and the bottom pressure for various shear modulus
to shear strength ratios. Reproduced from (24) by permission of
Crane, Russak & Company, Inc.
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133d'iN3W3DVldSI0 IVINOZIHOH H e c co o C N 0 -H H H u o co si
r-l 01 01 u J= 01 CO T3 CO C si a
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426
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Lecture Notes on Coastal and Estuarine Studies Estuaririe
Cohesive Sediment Dynamics Vol. 14
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Springer-Verlag in June 1992.