elastic waves in layered media ix - elastic...the cubical dilatation t, defined as the limit...

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CHAPTER IX

ELASTIC WAVES IN LAYERED MEDIA

9.1 INTRODUCTION

Oceans are a vast, complex, optically opaque but acoustically transparent world which is being

gradually explored by today’s science and technology. Underwater sound is used as the premier

tool to determine the detailed characteristics of physical and biological bodies and the processes

in the ocean. The wide-range of acoustic frequencies and wavelengths together with the diverse

oceanographic phenomena that occur over full spectra of space and time scales, thus gives rise to

a number of interesting effects and opportunities.

9.2 SOUND PROPAGATION IN LAYERED MEDIA

Various properties of sea such as density, pressure, salinity and temperature greatly influence the

propagation of sound in the medium. One of the most important factors is the nature of layers of

medium that are present in the ocean, namely sea-water and different kinds of sea-beds. Thus,

the propagation of sound can be considered as the propagation of elastic disturbances in layered

media, each layer being continuous, isotropic and constant in thickness.

For analytical study of sound propagation in ocean, different kinds of layered models may be

considered, for example 1. Two layered liquid half space, 2. Three layered liquid half space, 3.

Liquid layer on a solid bottom and 4. Solid layer over a solid bottom.

When sound interacts with the seafloor, the structure of the ocean bottom becomes important.

Ocean bottom sediments are often modeled as fluids which mean that they support only one type

of sound wave – a compressional wave. This is often a good approximation since the rigidity

(and hence the shear speed) of the sediment is usually considerably less than that of solids. In the

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latter case, which applies to ocean bed or the case where there is no sediment overlying the bed,

the medium must be modeled as elastic, which means it supports both dilatational and shear

waves. In reality, the media are visco-elastic, meaning they are also lossy.

A geoacoustic model is defined as a model of the real seafloor with emphasis on measured,

extrapolated and predicted values of those material properties important for the modeling of

sound transmission. In general, a geoacoustic model details the true thickness and properties of

sediment and rock layers within the seabed to the depth termed the effective penetration depth.

Clearly, the construction of a detailed geoacoustic model for a particular ocean is a tremendous

task. The approximate nature of information that is actually available is the primary limiting

factor on the accurate modeling of bottom-interacting sound transmission in the ocean. Many

problems of interest in geophysics and acoustics involve propagation of elastic disturbance in

layered half space.

In this chapter we shall consider the analysis of sound propagation in various cases consisting of

two or more layers. Love (1911) gave the first comprehensive treatment of the case of an elastic-

solid half space lying below a single solid layer. He calculated the dispersion of Rayleigh waves

and showed that a new surface wave having particle motion parallel to the surface and

perpendicular to the direction could exist under practical conditions. Stoneley (1937) investigated

the effect of the ocean on the transmission of Rayleigh waves, treating the effect of ocean bottom

as a solid half space.

9.2.1 TWO LAYERED LIQUID HALF SPACE

The dispersive waves observed by Ewing (1936) and Worzel et al (1948) in experiments with

sound explosion in shallow water were first interpreted by Pekeris (1948). He considered a

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problem of propagation of a disturbance in a two layered liquid half space. The results of Pekeris

for two liquid layers have often found applicable to studies on explosion sound transmission in

shallow water, implying absence of rigidity of ocean bottom materials. They also explained the

early experimental observation of Ewing (1936) that any accurate determination of the sound

velocity for horizontal transmission through the surface layer must be made at the highest

frequencies. Dobrin (1950) used the observed dispersion to deduce properties of the lagoon

bottom on Bikini atoll. He found that when the theory was applicable it could give useful

information on the bottom to a depth comparable to water depth.

9.2.2 THREE LAYERED LIQUID HALFSPACE

This problem was investigated by Pekeris (1950) and Press et el (1948). The case for which the

intermediate layer has a lower sound velocity than the first layer was investigated by Press

(1948). They were interested in the fact that in some areas the “water wave” consisted of a brief

burst of high frequency sound which did not show the dispersion and Airy phase normally found.

9.2.3 LIQUID LAYER ON A SOLID BOTTOM

Stoneley (1937) investigated the effect of ocean on the transmission of Rayleigh waves using

theory of plane waves. He calculated the phase and group velocities and concluded that the effect

of the water layer was unimportant for longer-period Rayleigh waves. Sezawa (1931) obtained an

approximate solution for the propagation of cylindrical waves, provided that the wave length was

great compared to the water depth, but he also neglected the shorter waves which are prominent

in many seismograms. Press et al (1948), in a study of microseisms, presented curves of phase

and group velocity for the first and second normal modes for plane waves in a liquid layer

superposed on a solid bottom.

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9.2.4 SOLID LAYER OVER SOLID HALF SPACE

Wave propagation in a solid layer of uniform thickness overlying a solid halfspace was first

studied by Bromwich (1898) for steady-state waves of length large compared with the layer

thickness. Love (1911) extended the work of Bromwich to include waves of length comparable

with or small compared with the layer thickness. This was an attempt to explain the duration and

complexity of earthquake waves.

9.3 FUNDAMENTAL EQUATIONS OF MOTIONS IN ELASTIC MEDIA

9.3.1 BASIC EQUATIONS IN ELASTIC MEDIA

When a deformable body undergoes a change in configuration due to application of a system of

forces, the body is said to be strained. Within the body, any point P with space-fixed rectangular

coordinates ( , , )x y z is the displaced to new position, the components of displacement being

, andu v w respectively. If Q is the neighboring point ( , , )x x y y z z of P, then the

displacement coordinates can be given by a Taylor expansion in the form

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

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

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

u u uu x y z

x y z

v v vv x y z

x y z

w w ww x y z

x y z

(9.1)

For the small strains associated with elastic wave, higher-order terms can be neglected. Then

introducing the expression

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1 1 1, ,

2 2 2

1 1 1, ,

2 2 2

x y z

xy yz zx

w v u w v u

y z z x x y

v u w v u we e e

x y y z z x

(9.2)

we may write the displacement components 9.1 in the following form

( ) ( ) .........................

( ) ( ) .........................

( ) ( ) .........................

y z xx xy xz

z x yx yy yz

x y zx zy zz

u z y e x e y e z

v x z e x e y e z

w y x e x e y e z

(9.3)

The first terms of these expressions are the components of the displacement of the point P. The

term in the parenthesis corresponds to a pure rotation of a volume element and the terms in the

second parenthesis are associated with deformation or strain of the element. The array

xx xy xz

yx yy yz

zyzx zz

e e e

e e e

ee e

(9.4)

represents the symmetric strain tensor at P, since , andxy yx yz zy zx xze e e e e e . The three

components , andxx yy zz

u v we e e

x y z

represent simple extensions parallel to the , ,x y z

axes and the other three expressions , ,xy yz zxe e e are the shear components of strain, which may-

be shown to be equal to half the angular changes in the , ,xy yz zx planes, respectively, of an

originally orthogonal volume element. It is also shown that in the theory of elasticity that there is

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a particular set of orthogonal axes through P for which the shear component vanish. These axes

are known as the principal axes of the strain. The corresponding values of , ,xy yz zxe e e are the

principal extensions which completely determine the deformation at P. Thus the deformation at

any point may be specified by three mutually perpendicular extensions. It is also known that the

sum xy yz zxe e e is independent of the choice of the orthogonal coordinate system.

The cubical dilatation , defined as the limit approached by the ratio of increase in volume to the

initial volume when the dimension , ,x y z approaches to zero, is

( )( )( )lim

xx yy zzx e x y e y z e z

x y z

xx yy zz

u v we e e

x y z

(9.5)

neglecting the higher order terms. The results hold for any cartesian coordinate system because

of the invariance of the sum.

Forces acting on an element of area S separating two small portions of a body are, in general ,

equivalent to a resultant force or traction R upon the element and a couple C as shown in Fig

9.1. As s goes to zero, the limit of the ratio of traction upon S to the area S is finite and

defines the stress. The ratio of the couple to S , involving an additional dimension of length may

be neglected. For a complete specification of the stress at P, it is necessary to give the traction at

P acting upon all planes passing through the point. However, all these tractions can be reduced to

component traction across planes parallel to the coordinate planes. Across each of these planes

the tractions may be resolved into three components parallel to the axes. However, all these

tractions may be reduced to component tractions across planes parallel to the coordinate planes.

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z

yxp

x

y P

C

R

S

yzp yxp

Fig 9.1 Traction R and C acting on element area S and

The stress components in plane normal to y axis.

Across each of these planes the tractions may be resolved into three components parallel to the

axes. This gives nine components of stress

xx xy xz

yx yy yz

zyzx zz

p p p

p p p

pp p

(9.6)

where the first subscripts represent a coordinate axis normal to a given plane and the second one

represents the axis to which the traction is parallel. The array 9.6 is a symmetrical tensor. This

may be proved by considering the equilibrium of small volume element within the medium with

sides of length , ,x y z parallel to the , ,x y z axes. Moments about axes through the centre

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y

x

z

x

y

z

xS

zyp

zxp

zzp

xy

zy

pp z

z

zzzz

pp z

z

zxzx

pp z

z

of mass arise from tractions corresponding to stresses , ,....xy yzp p . Moments of normal stresses

vanish, since the corresponding forces intersect the axes through the center of mass of the

infinitesimal element and moments of body forces are small quantities of higher order than those

of stresses. The equilibrium conditions require, therefore, that the shear components of strain,

three mutually perpendicular axes of principal stree may be found with respect to which the

shear components of stress vanish. Then the stress at a point is completely specified by three

principal stresses , ,xx yy zzp p p corresponding to these axes.

9.3.2 DERIVATION OF BASIC EQUATION

To derive the equations of motion we consider the tractions across the surface of a volume

element corresponding to the stress components 9.6 and the body forces , ,X Y Z which are

proportional to the mass in the volume element as shown in Fig 9.2.

Fig 9.2 Stress components in the faces zS of a volume element

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When the tractions are considered, the x components of the resultant force acting on an element,

e.g., produced by stresses in the faces normal to the , ,x y z axes, is (again neglecting higher

order terms)

xxxx xx x

yx

yx yx y

zxzx zx z

pp x p S

x

pp y p S

y

pp z p S

z

(9.7)

where , ,x y zS S S are the areas of the forces normal to the , ,x y z axes respectively. It

follows that the x component of force resulting from all the tractions is given by the three terms

yxxx zxpp p

x y zx y z

(9.8)

The equation of motion are obtained by adding all the forces and the corresponding inertia terms

2

2,........

d ux y z

dt , for each components where is the density of the medium.

2

2

2

2

2

2

yxxx zx

xy yy zy

yzxz zz

pp pd uX

dt x y z

p p pd vY

dt x y z

pp pd wZ

dt x y z

(9.9)

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9.3.3 THE EQUATION OF CONTINUITY

The equation expresses the condition that the mass of a given portion of matter is conserved. The

total outflow of mass from the elementary volume during the time t is -div v t

,

where v

is the velocity, whose components parallel to the , ,x y z axes are , ,u v w

. The loss of

mass during the same time is ( )t t . Equating these last two expression gives

div v = 0t

(9.10)

Another form of the equation is

div v = 0D

Dt

(9.11)

where the operator v gradD

Dt t

(9.12)

represents the “total or material” rate of change following the motion and t is the local rate

of change.

9.4 ELASTIC MEDIA

In the generalized form of Hooke’s law, it is assumed that each of the six components of tress is

a linear function of all the components of strain, and in general case 36 elastic constants appear

in the stress- strain relations.

9.4.1 ISOTROPIC ELASTIC SOLID

On account of the symmetry associated with an isotropic body, the number of elastic constants

degenerates to two, and the stress- strain relations may be written in the following manner, using

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the Lame’s constants and

2

2

2

xx xy

yy yz

zz zx

u u vp p

x y x

v v wp p

y z y

w w up p

y x z

(9.13)

We also could have written these equations using any two of the constants: Young Modulus E ,

Poisson’s ratio or the coefficient of incompressibility k .The relations between these elastic

constants are given by the equations

(1 )(1 2 ) 2(1 )

(3 2 ) 2;

2( ) 3

E E

E k

(9.14)

Using the Eq. 9.9 and 9.13 we can write the equations of motion in terms of displacements

, ,u v wof a point in an elastic solid

22

2

22

2

22

2

( )

( )

( )

uu X

t x

vv Y

t y

ww Z

t z

(9.15)

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We have replaced 2 2d dt by 2 2t , since it follows from 9.12 that the difference between

corresponding expressions involves second powers or products of components which are

assumed to be small. By neglecting these products, we liberalize our differential equations.

For many solids, and are nearly equal, and we will occasionally use the Poisson relations

as simplifications. This corresponds to5 1

and3 4

k . For incompressible medium ,

div v 0

or by 9.12 0D

Dt

9.4.2 IDEAL FLUID

If the rigidity vanishes, the medium is an ideal fluid. From Eq. 9.13 and 9.14 we find that

xx yy zzp p p p , where p , the value of the remaining independent component of the stress

tensor, is the hydrostatic or mean pressure. In liquids the incompressibility k is very large,

whereas it has only moderate values for gases. If a liquid is incompressible, and 0.5k .

The equations of small motion in an ideal fluid may be obtained from 9.15 with 0 .

9.5 IMPERFECTLY ELASTIC MEDIA

We shall also be concerned with the damping of elastic waves resulting from imperfections in

elasticity, particularly from “internal friction”. The effect of internal friction may be introduced

into the equations of motion by replacing an elastic constant such as by t in the

equation of motion. This is equivalent to starting that stress is a linear function of both the strain

and the time rate of change of strain. For simple harmonic motion, the time factor i te is used

and the effect of the internal friction is introduced by replacing by the complex rigidity of the

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medium (1 )i Q , where1 Q . In many cases Q may be treated as independent of

frequency to a sufficiently good approximation.

9.6 BOUNDARY CONDITIONS

If the medium to which the equations of motion are applied is bounded, some special conditions

must be added. These conditions express the behavior of stresses and displacement at the

boundaries. At free surface of a solid or liquid al stress components vanish. In the problems

which follow it will be assumed that solid elastic media are welded together at the surface of

contact, implying continuity of all stress and displacement components across the boundary. At a

solid-liquid interface slippage can occur and continuity of normal stresses and displacements

alone is required. Since the rigidity vanishes in the liquid, tangential stress in solid must vanish at

the interface.

9.7 REDUCTION TO WAVE EQUATION

The equation of motion of a fluid derived from Eq. 9.15 with 0 and k can be simplified

by introducing a velocity potential defined as follows

, ,u v wx y z

(9.16)

where , ,u v w

u v wt t t

(9.17)

If the body forces are neglected then Eq. 9.15 reduces to with the help of 9.17

, ,u v w

k k kt x t y t z

(9.18)

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Now writing 2 ,c k we easily see from Eq. 9.16 and Eq. 9.18 that

2 ( )c F tt

(9.19)

and 2

0

t

c dt (9.20)

where the additive function of t is omitted, being with significance.

From the definition of the mean pressure p k and Eq. 9.19 we have

pt

(9.21)

Differentiating Eq. 9.19 with respect to t and using Eq. 9.5 we obtain

2 2 2 2

2 2 2 2 2

1 u v w u v w

c t t t x y z x y z x y z

22

2 2

1

c t

(9.22)

In which small quantities of higher order have been neglected. This wave equation holds for

small disturbances propagating in an ideal fluid with velocity c under the assumption mentioned

above.

For displacements in a solid body, it is convenient to define a scalar potential and a vector

potential 1 2 3( , , ) as follows

3 32 1 2 1, ,u v wx y z y z x z x y

(9.23)

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or in vector form 1 2 3( , , ) grad curl ( , , )s u v w (9.24)

By the definition of as given by 9.5 we obtain 2 (9.25)

In general, the equation of motion 9.15 represent the propagation of a disturbance which involves

both equivoluminal ( 0 ) and irrotational ( 0 ) motion where div ( , , )s u v w and

1curl ( , , )

2s u v w .However by introduction of two types of velocity potentials and i ,

separate wave equations are obtained for these two types of motion. Assuming that the body

forces are neglected, we can write the first Eq. 9.15 in the following form

2 22

3 2

2 2 2

2 2 2 2

3 2( )

x t y t z t

x x y z

It is easy to see that this equation and the two others from 9.15 written in a similar form will be

satisfied if the functions and i are solutions of the equations

22

2 2

22

2 2

1

1, 1, 2, 3

d

ii

s

c t

ic t

(9.26)

where , ,d sc c

(9.27)

The wave equations 9.22 indicate that two types of disturbances with velocities dc and sc may be

propagated through an elastic solid.

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9.8 PROPAGATION OF WAVE IN SHALLOW WATER

9.8.1 BASIC INFORMATION

Acoustic waves in shallow water ocean may interact repeatedly with the seabed along the

propagation path. Theoretical analysis of the effect of sediment layers on acoustic propagation in

the ocean have been presented by Vidmar (1980a, b) and Harrison and Cousins (1985). Harrison

and Cousins (1985) discuss a number of different sediment and substrate configurations with

regard to relative dilatational and shear wave speeds. They state that in the case of very low

shear-speed sediment and a high-speed substrate, one would expect a resonance in the sediment

shear wave at a frequency corresponding to a layer thickness of one quarter wavelength. The

theory of unattenuated surface-wave propagation in a system consisting of a fluid layer of finite

thickness overlying a semi-infinite elastic solid has been the subject of numerous investigations

in the geophysical literature. The fluid-over-solid system is of considerable geophysical interest

since it represents a satisfactory approximation to a number of problems, such as the propagation

of Rayleigh waves over oceanic areas, of explosion generated surface waves in shallow waters,

possibly of long-period sound waves in the atmosphere, etc. The work evolved a series of

formulas which helped to clarify the computations and on the basis of which it was found

possible to compute a large number of cases. Special emphasis is given to calculate the

penetration depth and cut off frequency.

9.8.2 THEORY

We recapitulate the well established normal mode theory of the problem of finite water column

overlying homogeneous, isotropic elastic half-space for completeness. The finite water column

extends from free surface ( 0)z to seabed ( )z H and the solid, elastic half-space extends

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Z= 0

Z= H

Liquid Layer

, ,c

Solid Layer

, , , ,d s s d sc c

z

from seabed (𝑧 = 𝐻) to z . The density of water of the water column is and that of semi-

infinite solid layer is s . The sound speed in water is c. The dilatational wave speed in solid layer

is dc and the velocity of shear wave or distortional wave in the solid layer is sc . The elastic

modulus of the solid layer is E and its Poisson's ratio is.

Fig 9.3 Shallow Water wave guide

We introduce a displacement potential for the water column and two displacement potentials

d and s for the underlying solid half space. The potentials are solutions of the following wave

equations:

2 2 2 2 20 ( / )k k c in 0 z H (9.28)

2 2 2 2 2 2 2 2 2 20; 0 ( / ; / ) in d d d s s s d d s sk k k c k c H z (9.29)

where ( 2 )f is the source frequency in rad/s and f is the source frequency in Hz.

In the cylindrical coordinate system,

22

2

1r

r r r z

(9.30)

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In the cylindrical wave equation the wave equations .9.28 and 9.29 take the following form

2 22

2 2

10 ( or or )d sk

r r r z

(9.31)

The expressions of velocities in water and solid media are given by

, (0 )u w z Hr z

(9.32a)

21

, ( )d s d ss su w r H z

r r z z r r r

(9.32b)

The expressions for pressure (p) in the water column, normal stress zz and the shear stress rz in

solid layer are given by

2 2, 2 ands s szz s d s rz s

w u wp

z z r

(9.33a)

where2 2 2( ) ands s s d s s dc c c (9.34b)

Using separation of variable to solve Eq. 9.31, we choose

( , ) ( ) ( ); ( , ) ( ) ( ); ( , ) ( ) ( )d d d s s sr z R r Z z r z R r Z z r z R r Z z . (9.35)

Substituting Eq. 9.35 in Eq. 9.31 for , we have

2 22 2 2 2 2

2 2

10 and 0 ( )r r

R R Zk R Z k k

r r r z

The solutions of which, considering the finite water column which will admit sinusoidal depth

modes, are

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1 0 2 0 3 0 4 5( ) ( ) ( ) ( ) and ( ) cos sinr r rR r C J k r C Y K r C H k r Z z C z C z

Hence, by Eq. 9.35 0 1 2( , ) ( )( cos sin )rr z H k r D z D z (9.36)

Substituting Eq. 9.35 in Eq. 9.31 for andd s , we have

2 22 2 2 2 2

2 2

2 22 2 2 2 2

2 2

10 and 0 ( )

10 and 0 ( )

d d dr d d d d d r

s s sr s s s s s r

R R Zk R Z k k

r r r z

R R Zk R Z k k

r r r z

The solutions of which, considering the semi-infinite solid half-space which will admit

exponentially decaying depth modes such that when , and 0s dz , are

6 0 7

8 0 9

( ) ( ); ( )

( ) ( ) ; ( )

d

s

i z

d r d

i z

s r s

R r C H k r Z z C e

R r C H k r Z z C e

so that3 0 4 0( , ) ( ) ; ( , ) ( )d si z i z

d r s rr z D H k r e r z D H k r e (9.37)

Summarizing, we have

0 1 2 3 0 4 0( , ) ( )( cos sin ); ( , ) ( ) ; ( , ) ( )d si z i z

r d r s rr z H k r D z D z r z D H k r e r z D H k r e (9.38)

where , , andd s are defined by

Case I: ,d sc c c

2 2 2 2 2 2; ; ; / ; / ; /r d d r s s r d d s sk k k k k k k c k c k c (9.39a)

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Case II: s dc c c

2 2 2 2 2 2; ; ; / ; / ; /r d d r s r s d d s sk k k k k k k c k c k c (9.39b)

From Eqs. 9.38 and 9.39, it is clear that must be real so that depth modes in water column are

sinusoidal function and d and s must be positive and imaginary so that depth modes in solid

decay with increasing depth z , hence for evanescent mode

max ( , ) ,

min( , )

s d r d s

d r s s d

k k k k c c c

k k k k c c c

(9.40)

Boundary Conditions

The boundary conditions for the problem in Fig. 9.1 are as follows:

I. At 0z , the boundary is free surface. Hence, it is termed as pressure release and the stress

will be zero. This is used in fact that 0 at 0z (9.41)

II At z , the stresses vanish. This condition is used to define the dilational and shear

potentials as exponentially decaying functions along the depth. (see Eq. 9.37).

III At z H , the boundary is an interface between the liquid layer and the solid half space.

Therefore, the particle displacements due to wave propagation would be same at the interface to

ensure continuity. It is represented as sw w at z H (9.42)

IV At z H the normal stress components will be continuous between the liquid layers and solid

halfspace therefore, atzz zzLiquid Solidz H (9.43)

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and the shear stress component in the solid medium will be zero because the fluid is inviscid.

Hence, 0rz Solid at .z H (9.44)

where the stress are given by the following expressions

2

2 32 2

2 3

2 32

2

2 ( )

2 2

zz Liquid

d s szz d d sSolid

d s srz sSolid

k kz z z

kr z r z r

(9.45)

Applying the boundary condition 9.41 we have 1 0D and hence the velocity potentials are

reduce to 2 0 3 0 4 0( , ) ( )sin ; ( , ) ( ) ; ( , ) ( )d si z i z

r d r s rr z D H k r z r z D H k r e r z D H k r e

Applying the boundary condition 9.42 to 9.44 we have

2 2

2 3 4

2 2 2 2 2

2 3 4

2 2

2 3 4

[ cos ] [ ] [ ] 0

[ sin ] [ 2 ] [2 ( )] 0

[0] [2 ] [ 2 ]

d s

d s

d s

i H i H

d s s

i H i H

d d s s s

i H i H

d s s

D H D i e D k e

D H D k e D i k e

D D i e D k e

(9.46)

Defining 2 2 3 3 4 4, ,d si H i HD D D D e D D e

we have

2 2

2 3 4

2 2 2 2 2

2 3 4

2 2

2 3 4

[ cos ] [ ] [ ] 0

[ sin ] [ 2 ] [2 ( )] 0

[0] [2 ] [ 2 ]

d s s

d d s s s

d s s

D H D i D k

D H D k D i k

D D i D k

(9.47)

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Therefore, in matrix notation

2 2

2 2 2 2 2

2 2

cos( )

( ) sin( ) 2 2 ( ) 0

0 2 2

s

d s

s

d s

r s d s s s s

d s

H i k

F k H k i k

i k

(9.48)

The equation 9.48 is called the dispersion relation of the corresponding problem and in the given

range (Eq. 9.40) of rk Eq. 9.48 always have real value and the number of roots of this equation

determines the number of propagating modes.

9.8.3 STRESS DISTRIBUTION

Solving the system of equation 9.46 we have

2 2 2( )

2 4 3 4

2and

2 cos 2s s di H is s s

d

k kD D e D D e

H i

(9.49)

We assume 2 2 2

( )

1 2

2and

2 cos 2s s di H is s s

d

k kA e A e

H i

(9.50)

and hence 2 4 1 3 4 2andD D A D D A (9.51)

The stress distribution for the liquid is given by

2 2 2

2 0 4 1 0Liquid( )sin( ) ( )sin( )zz r rD H k r z D A H k r z

Liquid 2

1

4 0

sin( )( )

zz

r

A zD H k r

(9.52)

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Liquid0rz (9.53)

The normal stress for the solid layer is given by

2 2 2 2

3 4 0Solid{ ( 2 ) 2 ( ) } ( )d s

d

i z i z

zz s d s s s s s rD k e iD k e H k r

2 2 2 2Solid2

4 0

( 2 ) 2 ( )( )

d s

d

zz i z i z

s d s s s s s

r

A k e i k eD H k r

(9.54)

and the shear stress is

2 2

3 4 0Solid{2 ( 2 ) } ( )d s

s

i z i z

rz s d s r riD e D k e k H k r

2 2Solid2

4 0

2 ( 2 )( )

d s

s

rz i z i z

d s s s

r r

iA e k eD k H k r

(9.55)

For the solid layer z H so the normalize stress distribution in the solid layer can be expressed

as a function of z where z z H .Now, substituting z z H in 9.52 and 9.53 we have

2 2

( ) ( ) ( )2 2 2 2Solid

4 0

2( 2 ) 2 ( )

( ) 2

s s d d s

d

szz i n n H i z H i z H

s d s s s s s

r d

kk e e i k e

D H k r i

2 2

2 2 2 2Solid

4 0

2( 2 ) 2 ( )

( ) 2

s s d s s

d

szz in H i z i H i z

s d s s s s s

r d

kk e e i k e e

D H k r i

(9.56)

And

2 2

( ) ( ) ( )2 2Solid

4 0

22 . ( 2 )

( ) 2

s s d d s

s

srz i H i z H i z H

d s s

r r d

ki e e k e

D k H k r i

190

2 2

2 2Solid

4 0

22 . ( 2 )

( ) 2

s s d s s

s

srz i H i z i z i H

d s s

r r d

ki e e k e e

D k H k r i

(9.57)

Thus the mode shapes of the stresses are obtained from Eqs. 9.52, 9.53 for liquid layer and from

Eqs. 9.56 and 9.57 for solid layer. A procedure similar to the above may be used to obtain the

mode shapes of any other response quantity such as displacement components ( , )u w .Hence, the

normalized forms of stress functions are

2

1 4 0Liquid

Liquid

1 2 4 0Solid

1 2 4 0Solid

sin( ) , Normalize factor D ( )

0

, Normalize factor D ( )

, Normalize factor D ( )

d s

d s

zz r

zz

i z i z

zz r

i z i z

rz r r

A z H k r

Pe P e H k r

Q e Q e k H k r

(9.58)

where

2 2 2 2

1 2 2

2 2

1 2 2

2 22

1 2

( 2 ) 2 ( )

2 ( 2 )

2

2 cos 2

s

s

s

ss s

in H

s d s d s s s s

in H

s d s s

sin H in Hs

d

P k A P i k e

Q iA Q k e

kkA e A e

H i

(9.59)

The roots of the equation 9.48 and the stress distribution of in the shallow waveguide can be

ohtained through the MATLAB function “Shallow_Acoustic_Propagation_Calc” , “Matrix_4_

Shallow_water”, “Bisection_Method_Shallow_Water” and “Sigmazz_and_Sigmarz_Calc” and

they are shown in the Table 9.1 to 9.4.

191

The MATLAB function “Shallow_Acoustic_Propagation_Calc” also calculates the Penetration

Depth in the solid layer which will be discussed in the latter section of this chapter.

Table 9.1: Shallow_Acoustic_Propagation_Calc

%***************************************************************

% Shallow Water Propagation %*************************************************************** function [max_kr,min_kr,PD, sigma_zz, sigma_rz] =

Shallow_Acoustic_Propagation_Calc(f,c,rho,rhos,H,E1,niu) % c ----> Speed of compressional wave in water in m/s. % cd ----> Speed of compressional wave in solid in m/s. % cs ----> Speed of distortional wave in solid in m/s. % rho ----> Density of the water in kg/m^3. % rhos ----> Density of the solid in kg/m^3. % H ----> Water depth in m. % E1 ----> Young Modulus of the solid in GPA. % niu ----> Poisson Ration of the solid. % index ----> index = 1: cd,cs > c; index = 2: cs<c<cd.

% max_kr ----> Max Root of the Dispertion Releation.

% min_kr ----> Min Root of the Dispertion Releation. % sigma_zz ----> [sigma_zz at max kr; sigma_zz at min kr].

% sigma_rz ----> [sigma_rz at max kr; sigma_rz at min kr].

% PD ----> [PD at Max_kr, PD at min_kr] Penetration Depth

global mu k2 kd2 ks2 lamda total_depth index E= E1*1e9;total_depth= H+1000;increment=1e-5;ci=complex(0,1); omega=2*pi*f; lamda = (niu*E)/((1+niu)*(1-2*niu));mu = E/(2*(1+niu)); cd= sqrt((lamda+ 2*mu)/rhos);cs= sqrt(mu/rhos); k=omega/c; kd=omega/cd ; ks=omega/cs; k2=k^2; kd2=kd^2; ks2=ks^2; if (cd> c && cs > c) index=1;lower_kr = max(kd, ks); upper_kr = k; end if (cs< c && cd > c) index=2;lower_kr= kd; upper_kr= min(k,ks); end

last_value= lower_kr+increment;first_value= upper_kr-increment; range_maxkr= first_value increment :last_value; range_minkr= last_value increment : first_value; % CALACULATING THE RANGE OF MAX Kr and Min_kr for i = 1: 1: (length(range_maxkr)-1) A= Matrix_4_Shallow_water(range_maxkr(i)); B= Matrix_4_Shallow_water(range_maxkr(i+1)); if A*B < 0 min_lim_maxKr= range_maxkr(i+1); max_lim_maxKr= range_maxkr(i); break; end end for i = 1: 1: (length(range_minkr)-1) A= Matrix_4_Shallow_water(range_minkr(i)); B= Matrix_4_Shallow_water(range_minkr(i+1)); if A*B < 0

192

min_lim_minKr= range_minkr(i); max_lim_minKr= range_minkr(i+1); break; end end

%----- Modification of Root By Bisection Method --------------- % Maximum and Minimum kr interval. kr_lower= [min_lim_maxKr,min_lim_minKr]; kr_upper= [max_lim_maxKr,max_lim_minKr];

%m= 1 ==>> Max Kr [First Mode]

%m= 2 ==>> Min Kr [Last Mode] for m= 1: 1: 2 LL = kr_lower(m); UL = kr_upper(m); mod_root(m)= Bisection_Method_Shallow_Water(LL,UL); end max_kr= mod_root(1);% Maximum value of Kr min_kr= mod_root(2);% Minimum value of Kr %------ Calculating The Stress value -------------------------- totaldepth= 1: 0.1: total_depth; water_depth= length(1:0.1: H); for i= 1: 1: length(totaldepth) depth= totaldepth(i); [max_sigmazz(i), max_sigmarz(i)]= ... Sigmazz_and_Sigmarz_Calc(max_kr,depth); end for i= 1: 1: length(totaldepth) depth= totaldepth(i); [min_sigmazz(i), min_sigmarz(i)]= ...

Sigmazz_and_Sigmarz_Calc (min_kr,depth); end %------------------ Maximum Stress Value ----------------------- bound= water_depth+1: 1: length(totaldepth); Maximum_max_sigmazz = max(abs(max_sigmazz(bound))); Maximum_min_sigmazz = max(abs(min_sigmazz(bound))); %------------------- Penitration Depth-------------------------- PD_max= 0; PD_min= 0; for i = water_depth+1: 1:(length(totaldepth)) A = abs(max_sigmazz(i)); if (A <= 0.005*Maximum_max_sigmazz) S= max(abs(max_sigmazz(i+1:length(totaldepth)))); if 0.005*Maximum_max_sigmazz >= S PD_max= totaldepth(i); break end else PD_max= H; end end for i = water_depth+1: 1:(length(totaldepth)) A = abs(min_sigmazz(i)); if (A <= 0.005*Maximum_min_sigmazz) S= max(abs(min_sigmazz(i+1:length(totaldepth)))); if 0.005*Maximum_min_sigmazz >= S PD_min= totaldepth(i); break end

193

else PD_min= H; end end PD_max= PD_max- H; PD_min= PD_min- H; NFzz_max= abs(real(max_sigmazz(water_depth))); NFzz_min= abs(real(min_sigmazz(water_depth)));

Value= max_sigmarz(water_depth: 1:(length(totaldepth)));

NFrz_max= max(abs(value); NFrz_min= max(abs((min_sigmarz(water_depth:

1:(length(totaldepth)))))); Normalize_max_sigmazz= max_sigmazz/NFzz_max; Normalize_min_sigmazz= min_sigmazz/NFzz_min; Normalize_max_sigmarz= max_sigmarz/NFrz_max; Normalize_min_sigmarz= min_sigmarz/NFrz_min; sigma_zz= [Normalize_max_sigmazz;Normalize_min_sigmazz]; sigma_rz= [Normalize_max_sigmarz;Normalize_min_sigmarz];

PD= [PD_max,PD_min]; %------------------ End of the Function -----------------------

Table 9.2 : Matrix_4_Shallow_water

%***************************************************************

% Matrix For Shallow Water [Eq. 9.48] %***************************************************************

function [A] = Matrix_4_Shallow_water(krr) global H ci omega mu rho k2 kd2 ks2 lamda index krr2= krr^2;eta =sqrt(k2 -krr2);etad=sqrt(kd2-krr2); if index== 1 etas=sqrt(ks2-krr2);end if index== 2 etas=sqrt(krr2-ks2);end X= sin(eta*H);Y= cos(eta*H); etad2= etad^2;etas2= etas^2; A(1,1)=eta*Y;

A(1,2)=-ci*etad;

A(1,3)=etas2- ks2; A(2,1)=rho*(omega^2)* sin(eta*H);

A(2,2)= -lamda*kd2 - 2*mu*etad2; A(2,3)=2*ci*mu*etas*(ks2- etas2); A(3,1)=0; A(3,2)=2*ci*etad; A(3,3)=ks2-2*etas2; A = det(A);

%------------------ End of the Function -----------------------

Table 9.3 : Bisection_Method_Shallow_Water

%***************************************************************

% Bisection Method %***************************************************************

function [mod_root]= Bisection_Method_Shallow_Water(LL,UL)

%LL ---> Lower Limit of the initial guess of kr.

%LL ---> Lower Limit of the initial guess of kr. i= 1;tol = 1e-11; left(1)= LL; right(1)= UL; middle(1)= (LL+ UL)/2; if LL ~= UL

194

while i>= 1 f_l(i)= Matrix_4_Shallow_water (left(i)); f_r(i)= Matrix_4_Shallow_water (right(i)); f_m(i)= Matrix_4_Shallow_water (middle(i)); if f_l(i)*f_m(i)< 0 left(i+1) = left(i);

right(i+1)= middle(i); elseif f_m(i)*f_r(i)< 0 left(i+1) = middle(i);

right(i+1)= right(i); end middle(i+1)= (right(i+1)+ left(i+1))/2; if abs(middle(i+1)-middle(i))<= tol mod_root= middle(i); i= -1; else i= i+1; end end elseif LL == UL mod_root(mm)= LL; end %------------------ End of the Function -----------------------

Table 9.4 : Sigmazz_and_Sigmarz_Calc

%***************************************************************

% Stress Function Calulation [Eq. 9.58 & 59] %***************************************************************

function [sigmazz, sigmarz]= Sigmazz_and_Sigmarz_Calc(krr,depth)

%Krr ---> Modified Kr value determined by the MATLAB

% function ‘Bisection_Method_Shallow_Water’.

%depth ---> Depth calculated from the water surface. global k2 kd2 ks2 ci omega mu lamda H rho index krr2=krr^2; eta =sqrt(k2 -krr2);etad=sqrt(kd2-krr2); if index== 1 etas=sqrt(ks2-krr2);end if index== 2 etas=sqrt(krr2-ks2);end etad2=etad^2; etas2= etas^2; A1= (((ks2)*exp(ci*etas*H)))/(2*eta* cos(eta*H)); A2= -((ks2- 2*etas2)*exp(ci*etas*H))/(2*ci*etad); P1= (-lamda*kd2-2*mu*etad2)*A2; P2= 2*ci*mu*(ks2-etas2)*etas*exp(ci*etas*H); Q1= 2*ci*A2*mu*etad; Q2= (ks2- 2*etas2)*mu*exp(ci*etas*H);

Hankelf= besselh(0,krr);

Hankelfdas= besselh(1,krr); if depth <= H sigmazz= -A1*Hankelf *rho*(omega^2)*sin(eta*depth); sigmarz= 0; else zbar= depth - H; sigmazz= Hankelf*(P1*exp(ci*etad*zbar)+ ...

P2*exp(ci*etas*zbar)); sigmarz= -krr* Hankelfdas *(Q1*exp(ci*etad*zbar)+ ... Q2*exp(ci*etas*zbar)); end

195

sigmazz= real(sigmazz); sigmarz= real(sigmarz); end %------------------ End of the Function -----------------------

9.9 RESULTS AND DISCUSSION

9.9.1 SOIL/ROCK PROPERTIES

The solid substratum plays an important role in the nature of propagation. Table 1 shows the

properties of various soils usually found in the ocean floor obtained for Ewing et al (ref 8).

Table 9.5: Elastic properties of various soil/rock

Sea Bed

Type s

(kg/m3)

E

(GPa) cd (m/s) cs (m/s) cd/cs

(GPa)

(GPa)

Sedimentary

soil 1 1000 6.94

0.251 2885.82 1666.13 1.732 2.80 2.80

0.45 5130.71 1546.97 3.317 21.54 2.39

Sedimentary

soil 2 3000 7.5

0.25 1732.05 1000 1.732 3.00 3.00

0.45 3079.40 928.45 3.317 23.27 2.6

Sedimentary

soil 3 2000 25.3125

0.252 3897.11 2250.00 1.732 10.12 10.12

0.45 6928.67 2089.07 3.317 78.56 8.73

196

Rock 1

(Granite) 2500 56.25

0.253 5196.15 3000.00 1.732 22.50 22.50

0.45 9238.22 2785.43 3.317 174.57 19.39

Rock 2

(Hard rock) 1100 74.25

0.254 9000.00 5196.15 1.732 29.70 29.70

0.45 16001.08 4824.50 3.317 230.43 25.60

Rock 3

(Basalt) 3000 151.875

0.255 7794.23 4500.00 1.732 60.75 60.75

0.45 13857.33 4178.14 3.317 471.34 52.37

* From Ewing et al. (1957) 1: Fig. 4-28. 2: Fig. 4-26. 3: Fig. 4-18 4: Fig. 4-27. 5: Fig. 4-19.

From Table 1, it is noted that the Young modulus of sea-beds are spread over a very wide range

i.e., from 25-220 GPa. When propagation of sound is modeled in oceanic environment is also

have an important factor. According to Hamilton (ref. 7) the Poisson ratio of sea-beds are in the

range of 0.22-0.49.

9.9.2 NUMERICAL VALIDATION OF THE MATLAB FUNCTION

The determinant equation 9.48 has been solved using Bisection method. To validate the

numerical approach, few problems involving propagation of sound waves in liquid layer on a

solid substratum discussed in Ewing et al has been chosen and validation is given in Table 9.6.

Table 9.6: Validation of Phase velocities

(31500 m/s , 1000 kg/m , 0.25c )

Granite seabed

2H c 1k 1c c 1 (ref.)c c

0.1 0.003504 1.7929 1.79

0.3 0.013548 1.3913 1.40

0.5 0.028542 1.1007 1.10

0.7 0.042243 1.0412 1.04

1.0 0.061954 1.0142 1.01

6.0 0.374885 1.0056 1.00

Basalt seabed

0.1 0.002310 2.7200 2.70

0.2 0.004810 2.6125 2.60

0.3 0.011488 1.6408 1.61

0.5 0.027624 1.1372 1.14

1.0 1.027112 1.0271 1.03

2.0 1.005218 1.0052 1.00

197

Sedimentary soil 3 seabed

0.06 0.0028 1.3473 1.35

0.10 0.0048 1.3219 1.32

0.20 0.0102 1.2326 1.23

0.30 0.0168 1.1223 1.12

0.40 0.0239 1.0510 1.05

0.50 0.0309 1.0155 1.02

0.06 0.0028 1.3473 1.35

9.9.3 STRESS DISTRIBUTION IN THE HALFSPACE

Seabed Type: Sedimentary Soil 1 ( 0.25 )

We choose the sea bed as described in Table 9.5 with 100 m water depth and source frequency

50 HZ. Then the total number of propagating modes is 3 and the normalized stress distribution

for 1st mode (max rk ) and 3 rd mode (min rk ) are shown in the Fig. 9.4a and 9.5a calculated by

using the MATLAB function given in the Table 9.1 to 9.4 and here 1.7321d sc c and

1.1108sc c .

Fig 9.4a Normalized Stress for 1st Mode

198

Fig 9.5a Normalized Stress for 3rd Mode

SEABED TYPE: SEDIMENTARY SOIL 2 ( 0.25 )



for 1st mode (max rk ) and 3 rd mode (min rk ) are shown in the Fig. 9.4b and 9.5b calculated by


0.6667sc c .

199

Fig 9.4b Normalized Stress for 1st Mode

Fig 9.5b Normalized Stress for 3rd Mode

SEABED TYPE: SEDIMENTARY SOIL 3 ( 0.25 )



for 1st mode (max rk ) and 5 th mode (min rk ) are shown in the Fig. 9.4c and 9.5c calculated by


1.5sc c .

200

Fig 9.4c Normalized Stress for 1st Mode

Fig 9.5c Normalized Stress for 5th Mode

SEABED TYPE: Rock 1 [Granite] ( 0.25 )



for 1st mode (max rk ) and 6 th mode (min rk ) are shown in the Fig. 9.4d and 9.5d calculated by

using the MATLAB function given in the Table 9.1 to 9.4 and here 1.7321d sc c and 2sc c .

201

Fig 9.4d Normalized Stress for 1st Mode

Fig 9.5d Normalized Stress for 6th Mode

SEABED TYPE: Rock 2 [Hard Rock] ( 0.25 )



for 1st mode (max rk ) and 6 th mode (min rk ) are shown in the Fig. 9.4e and 9.5e calculated by


3.4611sc c .

202

Fig 9.4e Normalized Stress for 1st Mode

Fig 9.5e Normalized Stress for 6th Mode

SEABED TYPE: Rock 3 [Basalt] ( 0.25 )



for 1st mode (max rk ) and 7 th mode (min rk ) are shown in the Fig. 9.4f and 9.5f calculated by

using the MATLAB function given in the Table 9.1 to 9.4 and here 1.7321d sc c and 3sc c .

203

Fig 9.4f Normalized Stress for 1st Mode

Fig 9.5f Normalized Stress for 7th Mode

9.9.4 EFFECTIVE PENETRATION DEPTH

Penetration Depth is a measure of how deep the sound can penetrate into a material. It is defined

as the depth at which the stress due to the propagation of sound inside the material falls about

0.005% of its original value at (or more properly, just beneath) the surface. Effective Penetration

Depth is the extent up to which the stress variations for 1st mode i.e max kr due to propagation of

sound are significant in the seabed. This parameter is significant because of its ability to

204

determine the importance of seabed during the modeling of geoacoustic. Here, the effective

penetration depth is calculated as the depth, 0z at which the stress becomes just 0.5% of the

maximum value of the stress in the solid medium. This being a good approximation gives us a

valuable estimate of the penetration depths. Mathematically if zP is the penetration depth then

( ) 0.005 maxzz zzz PSolid Solid

(9.60)

Table 9.7 shows a comparison of penetration depths at various source frequencies for different

solid half space for a water depth of 100m and Table 9.8 tabulates penetration depth obtained at

various source frequencies and for different water depth for the granite sea bed .

Table 9.7: Comparison of penetration depths for different soil/rock properties,

with a water column depth H = 100m.

Penetration depth (m)

Freq

(Hz)

Sedimentary

soil 1

Sedimentary

soil 2

Sedimentary

soil 3

Rock 1

Granite

Rock 2

Hard Rock

Rock 3

Basalt

0.25 0.45 0.25 0.45 0.25 0.45 0.25 0.45 0.25 0.45 0.25 0.45

10 -- -- 246

(1)

118

(1)

611

(1)

778

(1)

213

(2)

215

(2)

196

(2)

196

(2)

201

(2)

201

(2)

30 145

(1)

504

(1)

94

(1)

40

(2)

83

(3)

87

(3)

73

(3)

73

(3)

62

(5)

62

(5)

63

(5)

63

(5)

50 77

(3)

148

(1)

38

(2)

23

(5)

47

(5)

48

(4)

41

(6)

41

(6)

38

(6)

38

(6)

37

(7)

37

(7)

100 37

(5)

63

(3)

17

(6)

11

(11)

23

(10)

24

(9)

20

(11)

20

(11)

19

(13)

19

(13)

19

(13)

19

(12)

200 18

(11)

30

(6)

8

(12)

6

(22)

12

(20)

12

(18)

10

(23)

10

(22)

9

(25)

9

(25)

9

(25)

9

(25)

300 12

(17)

20

(9)

6

(19)

4

(34)

8

(30)

8

(28)

7

(35)

7

(34)

6

(38)

6

(38)

6

(38)

6

(37)

400 9

(23)

15

(13)

4

(26)

3

(46)

6

(40)

6

(37)

5

(46)

5

(45)

5

(51)

5

(51)

5

(50)

5

(50)

500 7

(29)

12

(16)

4

(32)

2

(57)

5

(50)

5

(46)

4

(58)

4

(56)

4

(64)

4

(63)

4

(63)

4

(62)

750 5

(43)

8

(24)

2

(49)

2

(86)

3

(74)

3

(69)

3

(87)

3

(84)

3

(96)

3

(95)

3

(94)

3

(93)

1000 3

(58)

6

(32)

2

(66)

1

(115)

2

(99)

2

(93)

2

(115)

2

(112)

2

(128)

2

(127)

2

(126)

2

(124)

205

Table 9.8: Penetration depth for different water column depths at given

source frequency for granite seabed ( 56.25 GPa, 0.25 E )

Freq

(Hz)

Penetration depth (m)

H = 50 m H = 100 m H = 200 m H = 300 m H = 400 m H = 500 m

10 302

(1)

213

(2)

202

(3)

219

(3)

210

(5)

206

(6)

50 48

(3)

73

(3)

40

(11)

40

(17)

40

(23)

40

(29)

100 21

(6)

41

(6)

20

(23)

20

(35)

20

(46)

20

(58)

200 11

(11)

20

(11)

10

(46)

10

(69)

10

(92)

10

(115)

300 7

(17)

10

(23)

7

(69)

7

(104)

7

(139)

7

(172)

400 5

(22)

7

(35)

5

(92)

5

(139)

5

(185)

4

(230)

500 4

(28)

5

(46)

4

(115)

4

(173)

3

(230)

0.1

(288)

600 3

(35)

3

(69)

3

(139)

3

(208)

0.1

(276)

0.1

(345)

700 3

(40)

3

(81)

3

(162)

0.1

(242)

0.1

(322)

0.1

(403)

800 3

(46)

3

(92)

3

(185)

0.1

(277)

0.1

(368)

0.1

(461)

900 2

(52)

2

(104)

2

(208)

0.1

(311)

0.1

(415)

0.1

(519)

1000 2

(58)

2

(115)

2

(231)

0.1

(345)

0.1

(461)

0.1

(576)

In both the table the values in the open bracket gives the number of roots of the Eq. 9.48 and the

penetration depth are rounded off to the nearest integer. From the Table 9.7 it is observed that if

the Young modulus is small then the Poisson ratio have significant role on Effective penetration

Depth but for the large value of Young Modulus, the Poisson ratio loses its significance. Fig 9.6a

and 9.6b shows the comparison of penetration depth on different seabed for 100 m water depth at

varying frequency at 0.25, 0.45 respectively and Fig 9.7 shows the comparison on varying

frequency for different water column depth for granite sea bed.

206

Fig 9.6a Comparison of Penetration Depth on different seabed for 0.25

Fig 9.6b Comparison of Penetration Depth on different seabed for 0.45

207

Fig 9.7 Comparison of Penetration Depth on different water depth

for granite seabed ( 0.25 )

9.9.5 CUT-OFF FREQUENCY

During the analysis, it was observed that for a given water-column depth, H, there is a particular

source frequency such that the stress components in the seabed due to propagation of sound

becomes negligibly small so that we can ignore it and consider it to be zero. This frequency is

called the cut-off frequency and same behaviors are observed for the frequency greater than the

cut-off. Hence it has a great importance for the modeling of the shallow water. Table 9.8 lists the

cut off frequencies for different seabed types at various water column depths.

From Table 9.9 it is observed that for shallow waters (i.e., water column depths of up to 200 m)

there is penetration for higher source frequencies. Thus it is imperative that we cannot ignore

seabed effects while modeling shallow water geoacoustic problem. Fig 9.8a and 9.8b shows the

comparison between the cut-off frequencies for different sea bed respectively for 0.25, 0.45

208

Table 9.9: Cut-off frequency obtained for all the four kinds of seabed type

Cut-off frequency (Hz)

H

(m)

Sedimentary

soil 1

Sedimentary

soil 2

Sedimentary

soil 3

Rock 1

Granite

Rock 2

Hard rock

Rock 3

Basalt

0.25 0.45 0.25 0.45 0.25 0.25 0.25 0.45 0.25 0.45 0.25 0.45

100 4090 7280 1600 1410 2390 2560 2053 2110 1860 1870 1890 1910

200 2060 3640 800 710 1200 1280 1030 1060 930 950 950 960

300 1370 2430 540 470 800 860 690 710 620 630 630 640

400 1030 1820 400 360 600 640 520 530 470 470 480 480

500 820 1460 310 290 480 520 420 430 380 380 390 390

600 690 1220 270 240 400 430 350 360 310 320 320 320

700 590 1100 230 210 350 370 300 310 270 270 270 280

800 520 910 200 180 300 320 260 270 240 240 240 240

900 460 810 180 160 270 290 230 240 220 220 220 220

1000 410 730 160 150 240 260 210 220 190 190 200 200

Fig 9.8a Comparison of Cut-off frequency on different seabed for 0.25

209

Fig 9.8b Comparison of Cut-off frequency on different seabed for 0.45

Also from the Table 9.8 it is clear that if the Young modulus is small then the Poisson ratio have

a significant role on cut-off frequency but for the large value of Young Modulus, the Poisson

ratio loses its significance.

References

1 : Frank Press and Maurice Ewing, A Theory of Micrroseisms, with Geologic

Applications,Trans. Am. Geophys. Union, 29:163-174(1948).

2 : Frank Press, Maurice Ewing and Ivan Tolstoy, The Airy Phase of Shallow-Focus Sub-

marine Earthquakes, Bull. Seism. Soc. Am., 40:111-148(1950).

3 : M. A. Biot, The Interaction of Rayleigh and Stoneley Waves in the Ocean Bottom, Bull.

Seism. Soc. Am., 42:81-93(1952).

210

4 : Ivan Tolstoy and Maurice Ewing, The T phase of Shallow-Focus Earthquakes, Bull.

Seism. Soc. Am., 40:25-51(1950).

5 : Maurice Ewing, Frank Press and J. L. Worzel, Further Study of the T Phase, Bull. Seism.

Soc. Am., 42:37-53(1952).

6 : J. L. Worzel and Maurice Ewing, Explosion Sound in Shallow Water, Mem. Geol. Soc.

Am., 27 (1948)

7 : Edwin L. Hamilton, Vp/Vs and Poisson ratios in marine sediments and rocks, J. Acoust.

Soc. Am.,62(4),1093-1101 (1979).

8 : Ewing, M., Jardetzky, W., and Press, F.: Elastic Waves in Layered Media, McGraw-Hill

Book Company, Inc., 1957.

elastic waves in layered media ix - elastic...the cubical dilatation t, defined as the limit...

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