DTIC ADA227341: A Presentation on Perturbation Modeling for Ocean Sound Propagation

NUSC Technical Document 77T*
7 October 1986

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Mark 0. Duston
Ghasi R.Verma

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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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11 TITLE (Include Security Classification)

A PRESENTATION ON PERTURBATION MODELING FOR OCEAN SOUND PROPAGATION

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Vermi. and David H. Wood

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FROM_TO J I 19C6 October 7

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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.

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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

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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

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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

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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.

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