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NUSC Technical Document 77T* 
7 October 1986 


9 


779? 

Ujc03 


AD'A 227 341 


OTIC file 


copy 




Mark 0. Duston 
Ghasi R.Verma 



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OCT 12199° 




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Preface 


The work reported In this document was completed under NUSC 
Project No. A92045, Or. Rolf Kasper, Principal Investigator. A 
portion of this research was completed while two of us (Duston and 
Verma) participated in the U.S. Navy-ASEE Summer Faculty Research 
Program at the Naval Underwater Systems Center, New London Laboratory, 
in conjunction with NUSC Independent Research Project A92045. 


Reviewed and Approved: 7 October 1986 

{/uiWvxkJj>r 

A W. A. Von Winkle 
Associate Technical Director 
C' for Technology 







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Naval Underwater 
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11 TITLE (Include Security Classification) 

A PRESENTATION ON PERTURBATION MODELING FOR OCEAN SOUND PROPAGATION 


12 PERSONAL AUTHOR(S) 


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Vermi. and David H. Wood 


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FROM_TO J I 19C6 October 7 



COSATI COOES 


GROUP SUB-GROUP 


18. SUBJECT TERMS (Condnoa on reverse if necessary and identify by block number) 

Wave Propagation Perturbation 
Acoustics 
Modelin 


19. ABSTRACT (Continue on reverse if necessary and identify by block number) 

This document is based on a presentation given at the 12th International Congress 
on Acoustics in Halifax, Nova Scotia, Canada, 16-18 July 1986. 

\ 

We assume that the speed of sound in the water and the bottom of the ocean is a 
function of only the depth, and not the range. We also assume that the ocean and its 
bottom eventually interface with a rigid halfspace. This problem can be solved by the 
method of normal modes, involving the eigenvalues and eigenfunctions of a depth 
dependent ordinary differential equation. Since the sound speed in this problem 
varies only a little from its average value, -we- exploit the fact that the eigenfunctions 
and eigenvalues are known when the sound speed is constant. Vie investigate the changes 
in these eigenvalues and eigenfunctions that result from changes in the depth dependent 
sound speed within the ocean and its bottom, using an algebric formulation of the effect 


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19. ABSTRACT (Cont'd.) 

of the perturbation. Another more recent approach to finding the changes in 
the eigenvalues and eigenfunctions is a transmutation approach. We show a 
method of approximating the kernal of an integral transform and use it to find 
the first order corrections to the eigenvalues and eigenfunctions.’ Finally 
we compare the results of these two approaches with the results of classical 
perturbation theory for the same problem. 


_ UNCLASSIFIED _ 

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TD 7793 



GRAPH 1 

We would like to present an overview of some applications of perturbation 
theory for ordinary differential equations to ocean sound propagation. We are 
concerned generally with the problem of calculating the normal modes, and 
specifically we are involved with finding methods that allow us to find these 
normal modes in a much more efficient manner than present methods. 

We examine three different approaches involving p. rturbation for the 
calculation of normal modes. 

S’ " 

/ 

( ; ; 



1 









TD 7793 





WE WANT TO SOLVE THE FOLLOWING PROBLEM: 



p rr (r,z) +|p r (r,z) + p^z) + k 2 n2(z) p(r,z) = 0, 
WITH p(r,0) = 0 AND p 2 (r,h) = 0. 


V 




GRAPH 2 

We are dealing at this point with a fairly simplified model of the ocean 
and make the following assumptions: 

1) uniform depth, 

2) ocean is an isotropic medium, 

3) sound speed a function of depth only, 

4) pressure p-0 at the surface, 

5) ocean and its bottom are eventually (at some depth h) 
underlaid by a rigid surface. 

Under these assumptions the excess pressure p satisfies the Helmholtz Equa¬ 
tion, 

P rr (r ’ 2) + r p r (r ’ z) + P zz (r ’ z) + ^ 2 n 2 (z)p(r,z)- 0, 
with the boundary conditions 

p(r,0) - 0 and p z (r,h) - 0. 

2 

Here the term n (z) reflects the depth dependent nature of the sound speed. 


2 







TD 7793 



GRAPH 3 

We solve In a standard manner by using the method of separation of vari¬ 
ables p(r,z) - 0(z)9(r). The depth dependent equation gives the normal modes 
which satisfy 

0"(z) + k 2 n 2 (z)0(z) - Av(z) 

with the boundary conditions expressed as 

V>(0) - 0 and 0' (h) - 0. 


3 










TD 7793 





THE INDEX OF REFRACTION 
n^z) = 1 + €s(z) 

CONTAINS A PERTURBATION es(z). 

THE CASE 6=0 REPRESENTS AN IDEALIZED OCEAN 
WITH CONSTANT SOUND SPEED. 




GRAPH 4 

2 

The quantity n (z) is the index of refraction and we represent it as 

9 

n (z) - 1 + fs(z), 

where the quantity «s(z) is considered a perturbation . We introduce the 
parameter « which reflects the strength of the perturbation.. We assume that 
the function s(z) is known for the case of interest. When e-0 we have re¬ 
covered the idealized ocean with constant sound speed. 


4 




TD 7793 



GRAPH 5 

It is well known that for the idealized ocean with constant sound speed 
that the normal modes satisfy 

^"(z) + k 2 .£(z) - 2<t>( z) 

and the boundary conditions we imposed are 
■MO) - 0 and 0' (h) - 0. 

It is well known that this problem has a complete set of solutions with 
eigenfunctions given by 

, , . [h . (2m -l)ff7. 

V Z) ” \J 2 sln ~2h- rn - 1, 2, ... 

and corresponding discrete eigenvalues given by 

These eigenfunctions have been L 2 normalized 


5 






TD 7793 





THE GENERAL PERTURBATION APPROACH 


\ 


FIND THE EIGENFUNCTIONS \p n AND THE EIGENVALUES 
\ n OF THE PERTURBED PROBLEM IN TERMS OF THE 
EIGENFUNCTIONS <t> n AND EIGENVALUES (L n OF THE 
IDEALIZED PROBLEM AND THE PERTURBATION es(z). 


V_———— -* 

GRAPH 6 

In a perturbation approach we find the eigenfunctions V> m and eigenvalues 
A m of the perturbed (depth dependent) in terms of the eigenfunctions and 
eigenvalues 2 m of an idealized problem (which we know) and the perturbation 
fs(z) (which we also know). Specifically we look for the changes or correc¬ 
tion; which must be made to the idealized eigenfunctions and eigenvalues. 


6 









ID 7793 



GRAPH 7 


The first approach we examine is the classical perturbation approach 
found in Titchmarsh. The eigenvalues and eigenfunctions of the perturbed 
problem are expended in power series of the parameter e 


The perturbed eigenvalue equals the idealized eigenvalue plus corrections, 


i_ + € 
m 


+ < 2 A< 2 > + ... . 


The perturbed eigenfunctions equals the idealized eigenfunctions plus correc¬ 
tions , 

V z > - + ^ m ( 1 ) (2) + e 2 V> ra ( 2 ) (z) + ... • 


However, we find the corrections to the perturbed eigenfunctions only as a 
Fourier series of the idealized eigenfunctions and we must find the fourier 
coefficients a in *-erms of the idealized eigenfunction, the idealized 
eigenvalues and the perturbation. 


7 







THE CLASSICAL RESULTS 


THE CORRECTIONS TO FIRST ORDER IN € ARE 

x d) 2k 2 r . v L_ (2m - 1)7rz] 2 j 

X " = IT J 0 S{2) L -2h-j dz - 

THE FOURIER COEFFICIENTS ARE 


(1) = ?k! y s(2) «i„ ( 2 - m --J)2[z ftin (2n^ 1 )7rz 


a mn “ ft 


AND 


a min ~ 



GRAPH 8 

In a fairly straight forward manner the explicit equations for the corrections 
may be derived. We exhibit the corrections to the first order in t The first 
order corection to the m c ^ eigenvalue is given by the formula 

,( 1 ) 2 k 2 r h , , r . ( 2 m -l)»zl 2 . 

A - — J 0 5<z T ln —2h —J dz ■ 

The first order correction to the eigenfunctions is given in terms of an in¬ 
finite fourier series whose coefficients are given by 


(1 2k ... (2m -1)7rz _. (2n -1 )ttz 

a --rr s(z) Sln - ou - sin-rT - - 

mn h J Q 2h 2h 


a (1) - 0 
mm 


8 








I'D 7793 



GRAPH 9 

The next approach we examine is a Galerkin type approach. For compactness 
of notation we define L an infinte diagonal matrix whose entries are the 
eigenvalues of the idealized problem, $(z) an infinite column vector whose 
entries are the eigenfunctions of the idealized problem, A an infinite 
diagonal matrix whose entries are the eigenvalues of the perturbrd problem and 
l'(z) an infinite column vector whose entries are the eigenfunctions of the 
perturbed problem. 


9 









TD 7793 



GRAPH 10 

We may now express the perturbed problem in terms of the vector equation 
AT + k k (l + fs(z))<l> - At 
2 

where k (1 + es(z)) is a scalar quantity and the boundary conditions are 
4(0) - 0 and (h) - 0, 

where the 0 is a zero vector. The idealized problem is represented by a corre¬ 
sponding vector equation. 

In the Galerkin approach we look for a constant linear transformation 
(constant matrix) that satisfies 

1 i - D<f> 

In fact, we can guarantee the existence of such a matrix. The entries of the 
mth row of the matrix D are the coefficients of the r th eigenfunction 0 ex¬ 
pressed as a Fourier series in the eigenfunctions of the idealized problem. 


10 







TD 7793 



GRAPH 11 

Substituting the transformation with a normalization constraint we sim¬ 
plify and obtain an infinite algebraic eigenproblem 

(L + eA) D C - D C A, 

where A is a matrix defined by 

A =■ f k 2 s(z) dz 

J 0 

and c denotes the transpose. This is really a standard algebraic eigenproblem 
of the type MX - AX where the columns of D C are taken to be the vectors X. It 
is important that A multiply D C on the right so that the m C column of D 1 " is 
multiplied by the m 5 ^ eigenvalue in A. In this approach it is not necessary 
that the parameter e be small. The matrix A has special structure (it is 
symmetric, and the sum of Hankel and Toeplitz matrices) and it is this special 
structure which can be exploited to solve the problem more efficiently than an 
arbitrary eigenproblem. 


11 









TD 7793 



GRAPH 12 

The last approach we examine is a transmutation approach. First we 
express the solution of the perturbed normal mode problem in terms of the in¬ 
tegral transform of the solutions of an idealized normal mode problem. We use 
a transform of the type given by 

V> m (z) - <t>( z) + [ K(z ,s)4>(s) ds 
m n J h 

where we must specify the kernel function K(z,s). In fact, we do not solve 
for this kernel analytically but instead expand in a power series in the 
parameter «. We have developed an explicit method to find the coeffient of the 
power series expansion of the kernel and thus can approximate the kernel func¬ 
tion. 


12 









TD 7793 



THE TRANSMUTATION RESULTS 


THE CORRECTIONS TO FIRST ORDER IN e ARE 


x n = C n + 6 


(2n-1)7T f 2 (II_ 

2h ' / TT°" l°> 


AND 

v n ( 2 ) = o n (z) + e 


o?’m (|<-*| =osM! 

■(\l K^frsl^s) ds - jj | \l K^z.sl^s) dsl0< n O) (z)dz) 


/ 


GRAPH 13 


We can express the perturbed eigenvalues to the first order in e as 

and the perturbed eigenfunctions as 


^n (z) “ ^n (z) 

+ ‘K i)<o >( t 1 • 

+ [ J K (1) (z,s)/ 0) (s) ds - 



Z 1 - n -( 2n *!)« «(°) /_\ 1 

• hJ C0S -2h- ®n (Z) dz J 

,s)9^\s) ds | dzj j-. 


While the correction to the eigenfunction may look complicated it is the whole 
correction. We have an explicit formula without having to resort to Fourier 
series expansion. 


13 










THE TRANSMUTATION RESULTS 

(Cont’d) 


THE CORRECTIONS TO FIRST ORDER IN e ARE DEFINED 
IN TERMS OF 

^(Q) = fjK^O.s) 0<°>(s)ds 


AND 


. 2 / 2 - s + 2h 

e K^z.s) = (j h 2 [n%) - 1]df + j h 2 [n^) - l}dr 


V. 


/ 


GRAPH 14 

The c 9R ections eigenvalues and eigenfunctions are expressed in 
terras of ^ ; (0) and K u; (z,s). The first constant is defined by the expres¬ 
sion 


^(O) 


-t 


K (1) (O,s)0 (O) (s) 

n 


ds, 


and the first order in epsilon term of the power series expansion of the 
kernel is given by 


«K (1) (z,s) 




(z+s)/2 
[n (D 
h 


1] df + 


I, 


(z-s)/2 + h 

[n ($•) - 1) df 
h 


We again have the perturbation type result where the corrections are explicit 
functions of 9, the unperturbed eigenfunction, i , the unperturbed eigenvalue 
and the expression (n 2 (z) -1], which is the perturbation. 







COMPARISON OF APPROACHES 



GRAPH 15 

In summary let us examine each approach. The classic approach gives ex¬ 
plicit formulas that are readily obtained for the corrections but depends on 
both power series expansion in < and infinite Fourier series expansion of the 
corrections for the eigenfunctions. The Galerkin approach is neither depen¬ 
dent on small « nor does it require a power series expansion in that 
parameter, however, the eigenfunctions are given in terms of a Fourier series 
and does not give explicit formulas for the corrections. Finally the trans- 
method gives explicit formulas without the need for the Fourier 
series expansion, but it still requires a power series expansion in t and the 
calculations required are somewhat more involved. 














Abraham, Phil 

Code 5132, Physical Acoustic Branch 
Naval Research Lab. 

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Ahluwalia, D. S. 

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Woods Hole Oceanographic Inst. 

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Department of Mathematical Sciences 
University of Delaware 
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Computer Sciences Department 
University of Maryland, Baltimore County 
Cantonvi1le, MD 21228 

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ODSI Defense Systems, Inc. 

N. Stonington Prof. Center 
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Dept of Elec Eng and 
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GE, Research and Development 

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Department of Ocean Engineering 
MIT 

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University of Delaware 
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University of Rhode Island 
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Department of Mathematical Sciences 
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Dept of Eng Sci and Appl Math 
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Electrical Engineering 
University of Rhode Island 
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Aldridge Lab of Appl Geophysics 

Columbia Univ 

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Mathematics Department 
University of Rhode Island 
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Electrical Engineering 
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Department of Electircal Engineering 
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Department of Mathematics 
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Ocean Engineering Department 
University of Rhode Island 
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Liu, P. 

Mathematics Department 
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Lu, I. T. 

Dept of Elec Eng and 
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Polytechnic University 
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Madych, Wolodymyr 
Department of Mathematics 
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Department of Mathematics 
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Westinghouse Electric Corp, 
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Defense Research Establishment Atlantic 
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School of Mathematics 
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Department of Mathematics 
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Mullikin, Thomas 
Code 41 IMA 

Office Naval Research 
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Department of Mathematics 
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Department of Mathematics 
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Georgia Inst of Tech 
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IBM Corporation 
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Department of Mathematics 
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Computer Science Dept. 

Yale University 

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Department of Mathematics 
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Mathematics Department 
University of Rhode Island 
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Sine, Robert 

Department of Mathematics 
University of Rhode Island 
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Department of Mathematical Sciences 
University of Delaware 
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Sach/Freeman Associates 
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Department of Computer and 
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SACLANT ASW Research Centre 

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Department of Elec, Engineering 
Princeton University 
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Department of Mathematical Sciences 
Rensselaer Polytechnic Institute 
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Sciences Applications, Inc. 

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Cntr for Appl Math & Sci Comp 

Department of Mathematics 

University of Mass 

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Stakgold, Ivar 
Chairman 

Department of Mathematical Sciences 
University of Delaware 
Newark, DE 19716 

Stepanishen, Peter 
Department of Ocean Engineering 
University of Rhode Island 
Kingston, RI 02881 

Sternberg, Robert 
Office Naval Research 
495 Summer Street 
Boston, MA 02210 

Stickler, David 
Mathematics Department 
Colorado School of Mines 
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Department of Mathematics 
University of Rhode Island 
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Aldridge Lab of Appl Geophysics 

Columbia Univ 

New York, NY 10027 

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Defence Research Establishment Pacific 
Forces Mail Office 
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Canada VOS 1 BO 

Trefethen, Lloyd N. 

Department of Mathematics 
Mass Inst of Tech 
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Tufts, Donald 

Department of Electrical Engineering 
University of Rhode Island 
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Turkel, Eli 

Tel Aviv University 

Math Science School 

Ramat Aviv 

Tel Aviv, Israel 







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Van Loan, Charles 
Department of Computer Science 
Corneli University 
Ithaca, NY 14853 

Verma, Ghasi R. 

Department of Mathematics 
University of Rhode Island 
Kingston, RI 02881 


Vichnevetsky, R 
Department Computer Sci. 
Rutqers Univ 
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Vignes, J. 

Universite P. et M. Curie 
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France 

Weinberg, H. 

Senior Technical Director 
ODSI Defense Systems, Inc. 
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