N S T I T U T E O F WA T E R A C O U S T I C S, I G N A L...

I N S T I T U T E O F W A T E R A C O U S T I C S,

S O N A R E N G I N E E R I N G A N DS I G N A L T H E O R Y

1

Underwater Acoustics and Sonar Signal Processing

Contents1 Fundamentals of Ocean Acoustics2 Sound Propagation Modeling3 Sonar Antenna Design4 Sonar Signal Processing5 Array Processing

Chapter 2 / Sound Propagation Modeling / Prof. Dr.-Ing. Dieter Kraus



2

2 Sound Propagation ModelingSound propagation in the ocean is mathematically formulated by the wave equation, whose parameters and boundary con-ditions are descriptive of the ocean environment. As summa-rized in the figure below, there are a variety of different tech-niques available for solving the wave equation (numerically) for evaluating sound propagation in the sea.

AbbreviationsFE: Finite Element PE: Parabolic EquationFD: Finite Difference FFP: Fast Field ProgramNM: Normal Mode RT: Ray Tracing




3

FD/FE

RangeIndependent

Wave Equation

NM

CoupledFFP

CoupledNM

AdiabaticNM RT PE

FFP

RangeDependent




4

2.1 The Wave EquationThe wave equation in an ideal fluid can be derived from hy-drodynamics and the adiabatic relation between pressure and density. The following figure is used to derive the wave equation by exploiting the equation for conservation of mass, the Euler’s equation and the adiabatic equation of state.

x

pg(x,t)

v(x,t)

x

x + dx

pg(x + dx,t)

v(x + dx,t)A




5

For deriving the following equations we define the total pres-sure and the total density as follows.

where pg, p0 , p, ρg, ρ0 and ρ denote the total pressure, static pressure, change in pressure, total density, static density and change in density, respectively.

Continuity EquationEmploying the figure above the equation for the conservation of mass can be expressed by

pg = p0 + p and ρg = ρ0 + ρ,

ρg (x + dx,t)Av(x + dx,t)− ρg (x,t)Av(x,t)Resultant mass stream

! "####### $#######= −Adx

∂ρg

∂tdensity variation%

Mass variation! "## $##




6

and with

reformulated to obtain the so-called continuity equation

Euler’s EquationUsing the figure above Newton’s 2nd law can be written as

and by exploiting

pg (x,t)A− pg (x + dx,t)ATotal Force, F

! "#### $####= ρg Adx

V%

m!"$

dvdta%

ρg (x + dx,t)v(x + dx,t)− ρg (x,t)v(x,t)dx

=∂(ρgv)∂x

∂(ρgv)∂x

= −∂ρg

∂t= − ∂ρ

∂t.




7

and

we obtain Euler’s equation

Adiabatic Equation of State

dv = ∂v

∂tdt + ∂v

∂xdx, i.e. dv

dt= ∂v∂t

+ ∂v∂x

dxdt

= ∂v∂t

+ v∂v∂x

pg (x,t)− pg (x + dx,t)dx

= −pg (x + dx,t)− pg (x,t)

dx= −

∂pg

∂x

−∂pg

∂x= − ∂p

∂x= ρg

∂v∂t

+ v∂v∂x

⎛⎝⎜

⎞⎠⎟

.

pg = p0

ρg

ρ0

⎛

⎝⎜

⎞

⎠⎟

κ

= p0 +∂pg

∂ρg ρg=ρ0

ρ + 12∂2 pg

∂ρg2

ρg=ρ0

ρ2 +…




8

where κ denotes the adiabatic exponent.For convenience we define

which turns out later to be the squared sound speed in an ideal fluid. For

the adiabatic equation of state becomes approximately

Since the time scale of oceanographic changes is much longer than the time scale of the acoustical propagation, we suppose that the material properties ρ0 and c2 are independent of time.

c2 =∂pg

∂ρg ρg=ρ0

=κp0

ρ0

ρg

ρ0

⎛

⎝⎜

⎞

⎠⎟

κ −1

ρg=ρ0

=κp0

ρ0

pg ≅ p0 + c2ρ, i.e. p ≅ c2ρ.

p≪ p0 and ρ≪ ρ0




9

Taking the partial derivative of the continuity equation with respect to t and of Euler’s equation with respect to x provides

and

respectively. For lower particle velocities v as well as

the terms

2

2 r r¶ ¶ ¶ ¶ ¶æ ö æ ö- = +ç ÷ ç ÷¶ ¶ ¶ ¶ ¶è ø è øg g

p v vvx x t x x

22

2

2

2 2

( ) ( ),

1

g g

gg

v vp c

t x x t t

v pvx t x t c t

r r r r

rr

¶ ¶æ ö æ ö¶ ¶ ¶= = - =ç ÷ ç ÷¶ ¶ ¶ ¶ ¶è ø è ø

¶æ ö¶ ¶ ¶ ¶æ ö= + = -ç ÷ç ÷¶ ¶ ¶ ¶ ¶è øè ø

p≪ p0 and ρ≪ ρ0




10

can be neglected. Thus, the former equations simplify to

and provide by equating the 1dimensional linear wave equation

which can be extended by straightforward argumentation to the 3dimensional case given by

2 2

2 2 21 andr r¶ ¶ ¶ ¶ ¶ ¶æ ö æ ö= - - =ç ÷ ç ÷¶ ¶ ¶ ¶ ¶ ¶è ø è ø

g gv p p v

x t c t x x t

andr

r¶æ ö¶ ¶ ¶æ ö

ç ÷ç ÷¶ ¶ ¶ ¶è øè øg

gvv v

x t x x

2 2

2 2 21¶ ¶

=¶ ¶p px c t

Δp = 1

c2

∂2 p∂t2 with Δ = ∂2

∂x2 +∂2

∂y2 +∂2

∂z2 .




11

Helmholtz EquationSuppose harmonic pressure oscillation, i.e.

we obtain

If P possesses spherical symmetry, i.e. P is only depending on R, the Laplacian in spherical coordinates simplifies to

Hence, the spherical wave solution of the Helmholtz equation is

ΔP + k 2P = 0 with k =ω c = 2π λ .

p(x, y, z,t) = P(x, y, z)exp( jω t),

2

22 .

R R R¶ ¶

D = +¶ ¶

P(R)= Aexp(− jkR)

Rwith R= (x− xS )2 + ( y− yS )2 + (z− zS )2 ,




12

where xS, yS and zS are the coordinates of an omnidirectional point source (pulsating sphere of small radius).

Another simple and important solution is provided by the plane wave

where kx, ky and kz are the wave numbers that satisfy

The wave vector k can also be expressed by

where φ and θ denote the azimuth and elevation, respectively.

k 2 = kTk = kx

2 + ky2 + kz

2 , k =ω c = 2π λ .

( )( , , ) exp ( ) ,x y zP x y z A j k x k y k z= - + +

( ) ( ), , cos cos ,sin cos ,sin ,T T

x y zk k k k j q j q q= =k




13

2.2 Homogeneous WaveguideSuppose the medium within infinitely extended boundaries is homogeneous.

For the given point source coordinates (0,zS) the pressure shall be determined at an arbitrary receiver location (rR, zR).

¥

water columnc = 1480 m/sρ = 1 g/cm3

air

z

r

D

Rr

Rz

Sz Source

Receiver

0Sr =

sediment




14

2.2.1 Image Source Approach

The wave field within a homogeneous waveguide can be interpreted as the

superposition of infinitely many spheri-cal waves that are reflected at the boun-daries.

0

•

D

2D + zS

2D−zS

rrR

z

zR

zS

−zS

L01

L02

L03 L04

θ04θ03

θ02

Sediment

AirR1

R2

ImageSurface

ImageBottom




15

As a first approximation, the sound pressure in the waveguide can be determined by superimposing the four contributions indicated in the figure above, i.e.

P(rR , zR ,ω ) = A(ω ) e− j k L01

L01

+ R1(θ02 ,ω ) e− j k L02

L02

+⎛

⎝⎜

+ R2(θ03,ω ) e− j k L03

L03

+ R1(θ04 ,ω )R2(θ04 ,ω ) e− j k L04

L04

⎞

⎠⎟

with L01 = rR2 + (zR − zS )2

L02 = rR2 + (zS + zR )2

L03 = rR2 + (2D − zS − zR )2

L04 = rR2 + (2D + zS − zR )2

and

θ02 = arctan (zS + zR ) / rR( )θ03 = arctan (2D − zS − zR ) / rR( )θ04 = arctan (2D + zS − zR ) / rR( ).




16

Continuation of the image source technique in multiples m = 1,2,… of groups of four contributions provides

with

P(rR , zR ,ω ) = A(ω ) R1m(θm1,ω )R2

m(θm1,ω ) e− j k Lm1

Lm1

⎛

⎝⎜m=0

∞

∑

+ R1m+1(θm2 ,ω )R2

m(θm2 ,ω ) e− j k Lm2

Lm2

+ R1m(θm3,ω )R2

m+1(θm3,ω ) e− j k Lm3

Lm3

+ R1m+1(θm4 ,ω )R2

m+1(θm4 ,ω ) e− j k Lm4

Lm4

⎞

⎠⎟

2 2 2 21 3

2 2 2 22 4

(2 ) (2 ( 1) )

(2 ) (2 ( 1) )

m R S R m R S R

m R S R m R S R

L r Dm z z L r D m z z

L r Dm z z L r D m z z

= + - + = + + - -

= + + + = + + + -




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and

Taking into account that the reflection coefficients at the ocean surface and bottom can be approximated by

the calculation of the sound pressure simplifies to

θm1=arctan (2Dm− zS + zR ) / rR( ) θm3=arctan (2D(m+1)− zS − zR ) / rR( )θm2 =arctan (2Dm+ zS + zR ) / rR( ) θm4 =arctan (2D(m+1)+ zS − zR ) / rR( )

1 water-air-interface1 water-hard bottom-interface

RR» -»

P(r, z,ω ) = A(ω ) (−1)m e− i k Lm1

Lm1

⎛

⎝⎜m=0

∞

∑ − e− i k Lm2

Lm2

+ e− i k Lm3

Lm3

− e− i k Lm4

Lm4

⎞

⎠⎟.




18

Assignment 6:

Develop a Matlab program for determining P(r,z,ω) for the following parameters.

Signal parameters§ Sinusoidal waveform§ Amplitude: A =1,§ Frequency: f =10 Hz, 100 Hz, 1 kHz, 10kHz and 100 kHz

Waveguide parameters§ Water depth: D = 20 m § Source location: rS = 0 m, zS = 5 m§ Receiver location: § Surface/Bottom Reflection: R1 = −1 (calm), R2 = 1 (hard bottom)§ Sound speed: c = 1480 m/s

Depict the pressure distribution P(r,z,ω) in colour coded two dimensional diagrams and interpret the results.


(rR , zR )T ∈ [0,500]× [0, D]



19

2.2.2 Normal Mode Approach

For cylinder symmetric sound propagation, i.e. p = p(r,z,t), the wave equation is given by

Let us suppose that p can be expressed byHence, insertion in the wave equation provides

and after some manipulations

∂2 p∂r 2 + 1

r∂p∂r

+ ∂2 p∂z2 = 1

c2

∂2 p∂t2 .

( ) ( ) ( ).µp f r g z h t

2 2 2

2 2 2 2( ) ( ) ( ) ( )( ) ( ) ( ) ( )d f g z h t df d g f r g z d hg z h t f r h t

dr r dr dz c dt+ + =

2 2 2

2 2 2 21 1 1 1 1 .( ) ( ) ( )

d f df d g d hf r dr r dr g z dz c h t dt

æ ö+ + =ç ÷

è øChapter 2 / Sound Propagation Modeling / Prof. Dr.-Ing. Dieter Kraus



20

For harmonic sources with h(t) = Ae jωt we obtain

For all values of r and z, the r-dependent and z-dependent term are equal to constants. With the separation constant for the radial and for the vertical term, the separated ordinary dif-ferential equations are

The first equation is a zero-order Bessel equation. Its solution can be written in terms of a zero order Hankel function, i.e.

1f (r)

d 2 fdr 2 + 1

rdfdr

⎛⎝⎜

⎞⎠⎟

r -dependent term! "### $###

+ 1g(z)

d 2gdz2

z-dependent term! "# $#

= −ω2

c2 = −k 2.

2zk-

2rk-

2 22 2 2 2 2

2 21 0 and 0 with .r z r z

d f df d gk f k g k k kdr r dr dz

+ + = + = = +




whereJ0 = zero-order Bessel function of the 1st kind, (besselj(…)),Y0 = zero-order Bessel function of the 2nd kind, (bessely(…)),

also known as zero-order Neumann function N0,= zero-order Hankel function of the 1st kind, (besselh(…)),= zero-order Hankel function of the 2nd kind, (besselh(…)),

both are also known as zero-order Bessel function of the 3rd kind.

The asymptotic form of the Hankel function for is

21

(1)0 0 0(2)0 0 0

( ) ( ) ( )( )

( ) ( ) ( )r r r

r r r

H k r J k r jY k rf r

H k r J k r jY k rì = +

= í= -î

H0

(1) (krr)≅ 2π krr

ej krr − π

4⎛⎝⎜

⎞⎠⎟ and H0

(2) (krr)≅ 2π krr

e− j krr − π

4⎛⎝⎜

⎞⎠⎟ ,

(1)0H(2)0H


krr →∞



22

where and represent the converging and diverging cylindrical waves, respectively.The second equation represents an ordinary linear differential equation, where g has to satisfy the boundary conditions

Step I: (Elementary Solution)

With the set of independent solutions is given by

(1)0H

(2)0H

g(0) = 0 (R = −1)

g(D) = max (R = 1).

2

2 2 21,2

( ) ( )( ) ( ) 0 0

z z

z z z

g z e g z eg z k g z k jk

l ll

l l

¢¢= Þ =

¢¢ + = Þ + = Þ = ±

Re exp( jkz z){ } = cos(kz z) and Im exp( jkz z){ } = sin(kz z)

cos(kz z),sin(kz z){ }.Chapter 2 / Sound Propagation Modeling / Prof. Dr.-Ing. Dieter Kraus



23

Thus, the general solution of the ordinary linear differential equation can be expressed by

Step II: (Boundary Conditions)

The boundary conditions are satisfied for a discrete set of values of kz. Hence, we obtain

and

( ) cos( ) sin( ).z zg z A k z B k z= +

g(0) = 0⇒ A = 0g(D) = Bsin(kz D) = max ⇒ sin(kz D) = 1

⇒ kz D= (2m−1)π / 2, m=1,2,…

, (2 1) (eigenvalues)2z mk mDp

= -




24

for m = 1,2,… The solutions are called modes because they describe the natural ways in which the system vibrates.

The eigenvalues and are related by

( ) sin (2 1) (eigenfunctions)2m mg z B m zDpæ ö= -ç ÷

è ø

= = = =1 2 3 4m m m m

0

D

z

Pressure Amplitude [relative units]

k 2 = kr ,m

2 + kz ,m2 .2

,r mk2,z mk




25

Considering and to be the horizontal and vertical component of k, respectively, we can write

Consequently, we obtain

2,r mk

2,z mk

( ) ( ), ,( ) ( )cos ( ) and ( )sin ( ) .r m m z m mk k k kw w a w w a w= =

, 4

,

2( ) .r mj k r

mr m

f r ek r

p

p

æ ö- -ç ÷è ø@

,2 1( )r rk w e

,1 1( )r rk w e

,1z zk e

,2z zk e

1( )wk

1( )wk

,2 2( )r rk w e

,2z zk e 2( )wk

,1 2( )r rk w e

,1z zk e 2( )wk

2 1( )a w

1 1( )a w

2 2( )a w

1 2( )a w




26

Step III: (Initial Conditions)The last step is to add the fundamental solutions

in such a way that the initial condition

is satisfied. Substituting the sum into the initial condition gives

By multiplying each side of this equation with sin(kz,nz) and integrating from 0 to D, we obtain

,1 1

( ) sin( ) sin (2 1)2m z m m

m mg z B k z B m z

Dp¥ ¥

= =

æ ö= = -ç ÷è ø

å å

( ) ( )g z zf=

,1

( ) sin( ).m z mm

z B k zf¥

=

=å

2, ,0 0

( )sin( ) sin ( )2

D D

z n n z n nDz k z dz B k z dz Bf = =ò ò




27

due to the orthogonality property

Hence, the Bn are determined by

which are the Fourier coefficients of . For a source function given by

we have

, ,0

0sin( )sin( ) .

2D

z m z n

m nk z k z dz

D m n¹ì

= í =îò

,0

2 ( )sin( ) 1,2,...D

n z nB z k z dz nD

f= =ò

( ) ( )Sz z zf d= -

, ,0

2 2( )sin( ) sin( )D

n S z n z n SB z z k z dz k zD D

d= - =òChapter 2 / Sound Propagation Modeling / Prof. Dr.-Ing. Dieter Kraus

φ(z) (⇒ uniqueness)



28

and accordingly

Finally, for the boundary and initial condition given above the solution of the wave equation can be expressed by

, ,2( ) sin( )sin( ).m z m S z mg z k z k zD

=

p(r, z,t) = h(t) gm(z) fm(r)m=1

∞

∑

= Ae jωt 2D

sin(kz ,mzS )m=1

∞

∑ sin(kz ,mz) 2π kr ,m r

e− j kr ,mr − π

4⎛⎝⎜

⎞⎠⎟

= AD

8πr

ej ω t+ π

4⎛⎝⎜

⎞⎠⎟ sin(kz ,mzS )sin(kz ,mz)e− j kr ,m r

kr ,mm=1

∞

∑ .




29

2.3 Inhomogeneous WaveguideIf the sound speed in the water column is not constant the me-dium/waveguide is called inhomogeneous.

However, one generally assumes that the waveguide is cylinder symmetric with regard to the source location and is either

§ range independent, i.e. the medium is horizontally stratified such that c = c(z) is only a function of depth z

or§ range dependent, i.e. the sound speed varies versus horizon-

tal range and depth such that c = c(r,z) is a function of ranger and depth z.




30

In the following range independent scenarios are assumed for simplicity.

1480 1500

c (m/s)

z

water columnρ = 1 g/cm3

air

z

r

D

Rr

Rz

Sz Source

Receiver

0Sr =

sediment




31

2.3.1 Ray-Tracing, zero order sound speed approximation

The sound speed profile c(z) is approximated by a staircase function.

c(z) ≅ !c(z)

!c(z)

( )c z

[m/s]c1z2z

=Nz D

0z

z

15001480




32

sinεn

sinεn+1

=cn

cn+1

n = 1,…, N

Snell’s Law:

r1 r

ε1ε1

ε2

ε3

ε2

ε3

ε4

ε4

ε5ε5ε5

c2 > c1

c1

c3 > c2

c4 < c3

c5 < c4

r2

r3 r40

zS

z1

z5 = D

zSediment

Air

z2

z3

z4




33

The ray trace can be determined by applying Snell’s law at each boundary layer.

At the n-th boundary layer (0 = surface,…,N = bottom) holds

With

we obtain

sinεn = acn.

sinε1

c1

=sinε2

c2

=…=sinεN

cN

= a = const.

cos x = 1− sin2 x and tan x = sin x

cos x,

2 2

2 2cos 1 and tan

1n

n n n

n

a ca ca c

e e= - =-




34

such that the horizontal position and the travel time of the ray at the l-th boundary layer can be determined by

respectively, where z0 = zS, zN = D and l = 1,…,N.

2.3.2 Ray-Tracing, first order sound speed approximation

For any Snell’s law

provides

( ) 11

1 1( , ) tan and ( , )

cos

l ln n

n n nn n n n

z zr l a z z T l ac

ee-

-= =

-= - =å å

sin ( ) ( ) sin ( ) ( )e e= =S Sz c z z c z a

ε(z) = arcsin c(z)

c(zS )sinε(zS )

⎛

⎝⎜⎞

⎠⎟,


z ∈[0, D]



35

where

has been exploited. sinε(z) = ac(z) with a = sinε(zS ) c(zS )

( )( ) arcsin sin ( )( )

e eæ ö

= ç ÷è ø

SS

c zz zc z

( )e Sz

r

Sz

z

0




36

Using again the identities

we obtain

and

Horizontal partitioning ofthe water column in thin layers of thickness Δz as indicated in the figure on the right side leads to

2 2 2cos ( ) 1 sin ( ) 1 ( )z z a c ze e= - = -

2cos 1 sin and tan sin cosx x x x x= - =

2 2tan ( ) sin ( ) cos ( ) ( ) 1 ( ) .z z z ac z a c ze e e= = -0 r

Δz

Dz

!

!




37

For the Riemann sum becomes a Riemann integral so that we can write

Correspondingly, the travel time

( )

1( , ) ( , ) tan ( ).

N z

S Sn

r z a r z a z z n ze=

= + D + Då

r(z,a) = r(zS ,a)+ tanε( ′z )d ′zzS

z

∫

= r(zS ,a)+ ac( ′z )

1− a2c( ′z )2d ′z

zS

z

∫ .

( )

1( , ) ( , )

( )cos ( )

N z

Sn S S

zT z a T z ac z n z z n ze=

D= +

+ D + DåChapter 2 / Sound Propagation Modeling / Prof. Dr.-Ing. Dieter Kraus

Δz → 0



38

results in

for . Now, supposing the velocity profile can be approximated piecewise by linear functions, i.e.

T (z,a) = T (zS ,a)+ 1c( ′z )cosε( ′z )

d ′zzS

z

∫

= T (zS ,a)+ 1

c( ′z ) 1− a2c( ′z )2d ′z

zS

z

∫

c(zi)zi

zi+1

( ) ( ) ( )i i ic z c z g z z= + -

z

c0 r

Dz

zi !

!zi+1


Δz → 0



39

the integrals

and

can be solved for analytically.

Substitution with leads to

( )( )22

( ) ( )( , ) ( , )

1 ( ) ( )i

zi i i

iz i i i

a c z g z zr z a r z a dz

a c z g z z

¢+ -¢= +

¢- + -ò

T (z,a)=T (zi,a)+ d ′z

c(zi)+ gi( ′z − zi)( ) 1− a2 c(zi)+ gi( ′z − zi)( )2zi

z

∫

z ∈[zi , zi+1]

( ) ( )i i iv c z g z z= + - dv = gi dz( ) ( )

2 2( )

1( , ) ( , )1

i i i

i

c z g z z

iic z

a vr z a r z a dvga v

+ -

= + =-ò




40

The subsequent reformulation shows that rays follow circular paths in case of linear depth dependent velocity profiles.

= r(zi ,a)− 1− a2v2 agic( zi )

c( zi )+gi ( z−zi )

= r(zi ,a)+ 1− a2c(zi )2 − 1− a2 c(zi )+ gi(z− zi )( )2⎧

⎨⎩

⎫⎬⎭

agi

r(z,a)− r(zi ,a)+1− a2c(zi )

2

agi

⎛

⎝⎜⎜

⎞

⎠⎟⎟= −

1− a2 c(zi )+ gi(z − zi )( )2

agi

r(z,a)− r(zi ,a)+1− a2c(zi )

2

agi

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

=1− a2 c(zi )+ gi(z − zi )( )2

a2gi2




41

Moreover, the expression for the travel time can be similarly derived by

22 2 2

2 2 2

2 22 2

2 2

1 ( ) ( ( ) ( )) 1( , ) ( , )

1 ( ) ( ) 1( , ) ( , )

i i i ii

i i i

i ii i

i i i

a c z c z g z zr z a r z aag g a g

a c z c zr z a r z a z zag g a g

æ öæ ö- + -ç ÷ç ÷- + + =ç ÷ç ÷è øè ø

æ öæ ö æ ö- æ öç ÷ç ÷- + + - - =ç ÷ç ÷ç ÷ç ÷ è øè øè øè ø

T (z,a) = T (zi ,a)+ 1

v 1− a2v2

1gi

dvc( zi )

c( zi )+gi ( z−zi )

∫

= T (zi ,a)− 1gi

ln 1+ 1− a2v2

av

⎛

⎝⎜

⎞

⎠⎟

c( zi )

c( zi )+g ( z−zi )

=




42

Since explicit expressions r(z,a), T(z,a) are available for lin-ear velocity profiles, computationally efficient and satisfac-tory accurate ray-path calculations can be carried out after piecewise linear approximation of the actual velocity profile.

( )( )

( ) ( )( )

2 2

22

2 2

2 2 2

1 1 ( )1( , ) ln( )

1 1 ( ) ( ' )ln

( ) ( )

( ) ( ) 1 1 ( )1( , ) ln .1 1 ( ( ) ( )) ( )

ii

i i

i i

i i

i i i i

ii i i i i

a c zT z a

g a c z

a c z g z za c z g z z

a c z g z z a c zT z a

g a c z g z z a c z

ì æ ö+ -ï ç ÷= + í ç ÷ï è øîüæ ö+ - + - ïç ÷- ýç ÷+ - ïè øþ

æ ö+ - + -ç ÷= + ç ÷+ - + -ç ÷è ø




43

Literature[1] Brekhovskikh, L.M.; Lysanov, Y.P.: Fundamentals of Ocean

Acoustics, Springer, 2003[2] Etter, P.C.: Underwater Acoustic Modeling, Spon Press, 2003[3] Jensen, F.B: Computational Ocean Acoustics, Springer, 2000[4] Lurton, X.: An Introduction to Underwater Acoustics, Springer, 2004[5] Medwin, H.;Clay C.S: Acoustical Oceanography, Academic Press,

1998[6] Tolstoy, I.; Clay C.S: Ocean Acoustics, AIP-Press, 1987[7] Urick, R.I: Principles of Underwater sound, McGraw Hill, 1983


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