An energy-conserving spectral solution

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An energy-conserving spectral solutionMichael D. Collinsa)

Naval Research Laboratory, Washington, D.C. 20375

Henrik SchmidtMassachusetts Institute of Technology, Cambridge, Massachusetts 02139

William L. SiegmannRensselaer Polytechnic Institute, Troy, New York 12180

~Received 10 June 1999; accepted for publication 15 December 1999!

An energy-conserving spectral solution is derived and tested. A range-dependent medium isapproximated by a sequence of range-independent regions. In each region, the acoustic field isrepresented in terms of the horizontal wave-number spectrum. A condition corresponding to energyconservation is derived for the vertical interfaces between regions. The accuracy of the approach isdemonstrated for problems involving sloping ocean bottoms. The energy-conserving spectralsolution is less efficient than the energy-conserving parabolic equation solution, but might besuitable for generalization to problems involving elastic bottoms. ©2000 Acoustical Society ofAmerica.@S0001-4966~00!00704-9#

PACS numbers: 43.30.Bp, 43.30.Dr, 43.30.Gv@SAC-B#

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INTRODUCTION

Several approaches have been developed for solrange-dependent propagation problems in ocean acousRange-dependent normal-mode1 and spectral2 solutions arebased on a local separation of variables. It is importancontinue to develop these approaches despite the fact thaparabolic equation method3 usually provides greater efficiency by avoiding the field decomposition. All threethese approaches are based on the approximation of a radependent medium by a sequence of range-independengions. Accurate solutions can be obtained for many probleby applying single-scattering4 or energy-conservation5,6 con-ditions at the vertical interfaces between regions. Althousome progress has been made for problems involving elasediments,7–11 no existing approach provides the level of acuracy and efficiency that has been achieved for probleinvolving fluid sediments. In this paper, we derive an enerconserving spectral solution for problems involving flusediments. It might be possible to extend this approach toelastic case. An interface condition for conserving energdescribed in Sec. I. The derivation of the energy-conservspectral solution is presented in Sec. II. Examples aresented in Sec. III.

I. ENERGY CONSERVATION

In this section, we discuss background material andrive the complete energy-conservation correction,8 which isaccurate for problems involving gradual range dependeand a wide spectrum of horizontal wave numbers. We win the frequency domain and consider the case of asource in plane geometry, where the rangex is the horizontaldistance from the source andz is the depth below the oceasurface. Minor modifications are required for the more re

a!Electronic mail: [email protected]

1964 J. Acoust. Soc. Am. 107 (4), April 2000 0001-4966/2000/107

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istic case of a point source in cylindrical geometry.12 Arange-dependent medium is approximated by a sequencrange-independent regions. In each region, the acoustic psurep satisfies the wave equation,

]2p

]x2 1Lp52d~x!d~z2z0!, ~1!

L5r]

]z

1

r

]

]z1k2, ~2!

wherer is the density,k is the wave number, andz0 is thesource depth. Away from the source, we factor the operain Eq. ~1! into a product of outgoing and incoming operatoto obtain

S ]

]x1 iL 1/2D S ]

]x2 iL 1/2D p50. ~3!

Assuming that outgoing energy dominates incoming enerwe obtain the parabolic equation,

]p

]x5 iL 1/2p. ~4!

To completely define the solution, it is necessaryspecify conditions at the vertical interfaces between regioSince the parabolic equation contains only one range dertive, it is not possible to conserve both pressure and partvelocity across the vertical interfaces. Accurate solutionsbe obtained for many problems by conserving the eneflux,

E5Im E r21p*]p

]xdz. ~5!

The direct application of Eq.~5! would lead to a nonlinearinterface condition. To obtain a linear condition, we appthe modal representation,13

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p~x,z!5(n

anfn~z!exp~ iknx!, ~6!

in each range-independent region, where thean are constantsand thenth modefn(z) and eigenvaluekn

2 satisfy the equa-tions,

Lfn5kn2fn , ~7!

E r21fmfn dz5dmn . ~8!

Substituting the normal-mode representation into Eq.~5!and using the properties of the modes, we obtain

E5ReE r21p* L1/2p dz5E ur21/2L1/4pu2dz, ~9!

and conclude that the energy flux can be conserved byservingr21/2L1/4p across the vertical interfaces.

II. THE SPECTRAL SOLUTION

In this section, we describe the energy-conserving sptral solution. The subscriptj is used to denote evaluation ithe j th range-independent region, which corresponds toxj

,x,xj 11 . We express the acoustic pressure in terms ofwave-number representation,14

pj~x,z!51

2p E2`

`

p j~h,z!exp~ ih~x2xj !!dh, ~10!

where p j is the wave-number spectrum. In the first regiothe spectrum satisfies

~L12h2!p152d~z2z0!. ~11!

We define the fieldsF j (x,z) andGj 11(x,z) by the equa-tions,

F j~x,z!5L j1/4pj

51

2p E2`

`

h1/2p j~h,z!exp~ ih~x2xj !!dh, ~12!

]2Gj 11

]x2 1L j 11Gj 1152d~x2xj 11!F j~xj 11 ,z!. ~13!

The spectral solution of Eq.~13! is

Gj 11~x,z!51

2p E2`

`

Gj 11~h,z!exp~ ih~x2xj 11!!dh,

~14!

~L j 112h2!Gj 1152F j~xj 11 ,z!. ~15!

Integrating Eq.~13! over an arbitrarily small intervaaboutx5xj 11 , we obtain

]Gj 11

]x~xj 11 ,z!5F j~xj 11 ,z!. ~16!

From Eq.~14!, we obtain

]Gj 11

]x5

i

2p E2`

`

hGj 11~h,z!exp~ ih~x2xj 11!!dh.

~17!

1965 J. Acoust. Soc. Am., Vol. 107, No. 4, April 2000


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Setting r j 1121/2F j 11(xj 11 ,z)5r j

21/2F j (xj 11 ,z) and applyingEqs.~12!, ~16!, and~17!, we obtain

ir j21/2hGj 115r j 11

21/2h1/2p j 11 . ~18!

From Eqs.~15! and ~18!, we obtain

p j 1152ir j 111/2 r j

21/2h1/2~L j 112h2!21F j~xj 11 ,z!. ~19!

The energy-conserving solution is generated using Eqs.~12!and ~19!.

III. IMPLEMENTATION AND EXAMPLES

To obtain a stable solution, it is necessary to chooseintegration contour carefully. We have obtained good resusing an integration contour that consists of five linear sments. The endpoints of the segments areh5 i e, h50, h5ha , h5ha2 id, h5hb2 id, and h5hb1 i e. The wavenumbersha andhb are selected so that the sharp peaks inspectrum lie between these values. The small parameteeand d are selected so that the integrand is negligibleIm(h).e and to avoid the peaks in the spectrum in the regha,Re(h),hb . For the examples, we use values ofha andhb that correspond to horizontal phase speeds of 1400

FIG. 1. Transmission loss for exampleA, which involves a 25-Hz source ina water column that decreases in depth with range. The dashed curverespond to a reference solution that was generated using the enconserving parabolic equation.~a! The energy-conserving spectral solutio~solid curve! is in agreement with the reference solution.~b! The pressure-conserving solution~solid curve! contains a negative amplitude error.

1965Collins et al.: An energy-conserving spectral solution


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and 1800 m/s. Reference solutions were generated usiparabolic equation implemented with the complete enerconservation correction.

Example A is identical to the wedge benchmaproblem15 with the exception that we consider a line sourin plane geometry. This problem involves a 25-Hz sourcez5100 m in a water column in whichc51500 m/s. Theocean depth decreases linearly from 200 m at the solocation to zero atx54 km. In the sediment,c51700 m/s,r51.5 g/cm3, and the attenuation is 0.5 dB/l. Transmissionloss for exampleA appears in Fig. 1. The energy-conservispectral solution is in agreement with the reference solutThe pressure-conserving solution contains a negative amtude error. ExampleB is identical to exampleA, with theexception that the ocean depth increases linearly to 400x54 km. Transmission loss for exampleB appears in Fig. 2.

FIG. 2. Transmission loss for exampleB, which involves a 25-Hz source ina water column that increases in depth with range. The dashed curvesrespond to a reference solution that was generated using the enconserving parabolic equation.~a! The energy-conserving spectral solutio~solid curve! is in agreement with the reference solution.~b! The pressure-conserving solution~solid curve! contains a positive amplitude error.

1966 J. Acoust. Soc. Am., Vol. 107, No. 4, April 2000


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The energy-conserving spectral solution is in agreement wthe reference solution. The pressure-conserving solution ctains a positive amplitude error.

IV. CONCLUSION

An energy-conserving spectral solution has been deriand tested. The complete energy-conservation correctioapplied at vertical interfaces in two steps. A quantity thatrelated to the pressure by the fourth root of the depth optor is first computed using a spectral integral. The forcifunction for the spectrum in the next range-independentgion is composed of this quantity plus factors of the squroots of density and horizontal wave number. The accurof the energy-conserving spectral solution was demonstrfor problems involving sloping ocean bottoms. Although thapproach is not as efficient as the energy-conserving pbolic equation, it might be easier to generalize to the elacase, which is an important unresolved problem.

ACKNOWLEDGMENT

This work was supported by the Office of Naval Rsearch.

1R. B. Evans, ‘‘A coupled mode solution for acoustic propagation inwaveguide with stepwise depth variations of a penetrable bottom,’Acoust. Soc. Am.74, 188–195~1983!.

2K. E. Gilbert and R. B. Evans, ‘‘A Green’s function method for one-wwave propagation in a range-dependent ocean environment,’’ inOceanSeismo-Acoustics, edited by T. Akal and J. M. Berkson~Plenum, NewYork, 1986!.

3F. B. Jensen, W. A. Kuperman, M. B. Porter, and H. Schmidt,Computa-tional Ocean Acoustics~American Institute of Physics, New York, 1994!,pp. 343–412.

4M. D. Collins and R. B. Evans, ‘‘A two-way parabolic equation for acoutic backscattering in the ocean,’’ J. Acoust. Soc. Am.91, 1357–1368~1992!.

5M. B. Porter, F. B. Jensen, and C. M. Ferla, ‘‘The problem of eneconservation in one-way models,’’ J. Acoust. Soc. Am.89, 1058–1067~1991!.

6M. D. Collins and E. K. Westwood, ‘‘A higher-order energy-conservinparabolic equation for range-dependent ocean depth, sound speeddensity,’’ J. Acoust. Soc. Am.89, 1068–1075~1991!.

7M. D. Collins, ‘‘A two-way parabolic equation for elastic media,’’ JAcoust. Soc. Am.93, 1815–1825~1993!.

8M. D. Collins, ‘‘An energy-conserving parabolic equation for elastic mdia,’’ J. Acoust. Soc. Am.94, 975–982~1993!.

9H. Schmidt, W. Seong, and J. T. Goh, ‘‘Spectral super-element approto range-dependent ocean acoustic modeling,’’ J. Acoust. Soc. Am.98,465–472~1995!.

10J. T. Goh and H. Schmidt, ‘‘A hybrid coupled wave-number integratiapproach to range-dependent seismoacoustic modeling,’’ J. Acoust.Am. 100, 1409–1420~1996!.

11M. D. Collins and W. L. Siegmann, ‘‘A complete energy conservaticorrection for the elastic parabolic equation,’’ J. Acoust. Soc. Am.105,687–692~1999!.

12Reference 3, pp. 272–277.13Reference 3, pp. 271–341.14Reference 3, pp. 203–269.15F. B. Jensen and C. M. Ferla, ‘‘Numerical solutions of range-depend

benchmark problems,’’ J. Acoust. Soc. Am.87, 1499–1510~1990!.

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1966Collins et al.: An energy-conserving spectral solution


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An energy-conserving spectral solution