# Full text of "DTIC ADA227341: A Presentation on Perturbation Modeling for Ocean Sound Propagation"

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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 5 OTIC ELECTEl OCT 12199° 5SJ ffiSSfK SS . wl _ dUtflbut'on IlmH**- Approve «a» pu wte rW * w S<‘ 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 SECURITY CLA PAuE la. REPORT SECURITY CLASSIFICATION UNCLASSIFIED 2a. SECURITY CLASSIFICATION AUTHORITY REPORT DOCUMENTATION PAGE lb. RESTRICTIVE MARKINGS 2b. OECLASSIFICATION / DOWNGRADING SCHEDULE 4 PERFORMING ORGANIZATION REPORT NUMBER(S) TD 7793 3 DISTRIBUTION .'AVAILABILITY OF REPORT Approved for public release; distribution unlimited 5. MONITORING ORGANIZATION REPORT NUMBER(S) 6a. NAME OF PERFORMING ORGANIZATION Naval Underwater Systems Center 6c AOORESS ( Cty; Stan, and ZIP Coda). New London Laboratory New London, CT 06320 3a. NAME OF FUNOING/ SPONSORING ORGANIZATION 3c AOORESS (City, Stan, and ZIP Coda) 11 TITLE (Include Security Classification) A PRESENTATION ON PERTURBATION MODELING FOR OCEAN SOUND PROPAGATION 12 PERSONAL AUTHOR(S) 6b. OFFICE SYMBOL I 7a. NAME OF MONITORING ORGANIZATION (If applicable) I 7b. AOORESS (Cty, Stan, and ZIP Coda) 8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER (If applicable) I 10. SOURCE OF FUNOING NUM8ERS PROGRAM PROJECT TASK ELEMENT NO. NO NO. 13a. TYPE OF REPORT Vermi. and David H. Wood 13b. TIME COVEREO 14. DATE OF REPORT (year. Month, Day) Il5 PAGE COUNT 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 21 ABSTRACT SECURITY CLASSIFICATION UNCLASSIFIED 20 DISTRIBUTION / AVAILABILITY OF ABSTRACT EuNCLASSIFIEDAJNUMtTED □ SAME AS RPT Q OTIC USERS 22* NAME OF RESPONSIBLE INOIVIOUAL |22b TELEPHONE (Include Area Code) 22c OFFICE SYMBOL David H. Wood_I (203) 440-4831 Code 3332 DO FORM 1473, 84 MAR 83 APR edition may be us*d until exhausted SECURITY CLASSIFICATION OF this PAGE All oth#r tditiom art obiolttt. 1 UNCLASSIFIED UNCLASSIFIED _ security classification of this page 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 _ SECURITY CLARIFICATION OF THIS PAGE 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. Washington, DC 20375 Ahluwalia, D. S. Chairman Mathematics Department New Jersey Institute of Technology Newark NJ 07102 Ames, W. F. Georgia Inst of Tech Atlanta, GA 30332 Andrushkiw, R. 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