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Numerical computation of the scattering frequencies for acoustic wave equations. (English) Zbl 0643.65080
The first step of the described method is to solve the time dependent wave equation $$u_{tt}-\Delta u+q(x)u=0$$ for any convenient nontrivial initial data. The second step is to fix x and calculate the exponents of the known asymptotic expansion $$u\sim \Sigma c_ jp_ j(x)\exp (is_ jt)$$ from the step one discrete numerical solutions evaluated at equally spaced times. This delicate numerical procedure, previously investigated, uses a classical idea of Prony which reduces the problem to solving two systems of linear equations and finding the zeros of a polynomial of degree n. Example calculations and a discussion of accuracy complete the paper.
Reviewer: W.Ames

##### MSC:
 65Z05 Applications to the sciences 76Q05 Hydro- and aero-acoustics 35Q99 Partial differential equations of mathematical physics and other areas of application 78A45 Diffraction, scattering 35L15 Initial value problems for second-order hyperbolic equations
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##### References:
 [1] Cooper, J.; Strauss, W., Indiana univ. math. J., 34, 33, (1985) [2] Cooper, J.; Perla-Menzala, G.; Strauss, W., Math. methods in appl. sci., 8, 576, (1986) [3] Goodhue, W., Trans. amer. math. SAC., 180, 337, (1973) [4] Ikawa, M., J. math. Kyoto univ., 23, 127, (1983) [5] Lax, P.; Phillips, R., Scattering theory, (1967), Academic Press New York · Zbl 0214.12002 [6] Lax, P.; Phillips, R., Commun. pure appl. math, 22, 737, (1969) [7] \scG. Majda, W. Strauss, and M. Wei, “Computation of exponentials in transient data,” submitted for publication. · Zbl 0946.78520 [8] Melrose, R., J. funct. anal., 53, 287, (1983) [9] Prony, R., J. l’ecole polytech. (Paris), 1, 2, 24, (1795), cahier [10] Richtmyer, R.; Morton, K., Difference methods for initial-value problems, (1967), Interscience New York · Zbl 0155.47502 [11] Strauss, W.; Majda, G.; Wei, M., Matemática aplicada e computacional, 6, 17, (1987) [12] \scM. Wei, Ph.D. thesis, Brown University, Providence, 1986 (unpublished). [13] Dolph, C.; Cio, S., IEEE trans. antennas prop., 28, 888, (1980) [14] Reinhardt, W., Ann. rev. phys. chem., 33, 223, (1982)
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