## A combined method for computing the field of the point source in a waveguide (plane-layered media).(Russian, English)Zbl 1130.35122

J. Math. Sci., New York 132, No. 1, 113-129 (2006); translation from Zap. Nauchn. Semin. POMI 308, 197-224, 255-256 (2004).
Summary: The two-dimensional problem of propagating the waves generated by a point source, in a depth plane-layered waveguide is investigated. The Dirichlet condition is given on the boundary of the waveguide and also two consistency conditions are defined on the interface between the fluid and the bottom. The velocity of acoustic waves is assumed to be dependent only on the depth of the waveguide. With the help of cumbersome transformations, the solution is represented as a sum of geometric-optical waves, normal waves, and a remainder. Sufficient conditions on the total amount of detached normal and geometric-optical waves are obtained. The remainder is expressed by a simple formula.

### MSC:

 35Q60 PDEs in connection with optics and electromagnetic theory 76Q05 Hydro- and aero-acoustics 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 78A05 Geometric optics 78A50 Antennas, waveguides in optics and electromagnetic theory
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### References:

 [1] V. S. Buldyrev, V. S. Lanin, and Z. A. Yanson, ”Calculation of the field on the axis of a symmetric waveguide,” in: Problems in the Dynamic Theory of Propagation of Seismic Waves (1974), pp. 1–84. [2] V. E. Grikurov and M. M. Popov, ”Summation of Gaussian beams in a surface waveguide,” Wave Motion, 5, 225–233 (1983). · Zbl 0511.73020 [3] A. P. Katchalov and M. M. Popov, ”An application of the method of Gaussian beams summation to calculation of high-frequency wave fields,” Dokl. Akad. Nauk SSSR, 258, No.5, 1097–1100 (1981). [4] A. P. Katchalov and M. M. Popov, ”An application of the method of Gaussian beams in isotropic elasticity theory,” LOMI P-9-82, Leningrad (1982). [5] A. S. Krjukovskii, D. S. Lukin, and E. A. Palkin, ”Numerical comparison of two asymptotic methods of studying wave diffraction problems in smooth inhomogeneous media,” Izv. Vyssh. Uchebn. Zaved. (VUSov), Radiofizika, 29, No.1, 79–88 (1986). [6] V. E. Grikurov, ”The effect of overlapping near-by-caustics regions in a surface waveguide and a generalization of the ray method,” Izv. Vyssh. Uchebn. Zaved. (VUSov), Radiofizika, 23, No.9, 1038 (1980). [7] V. M. Kudryashov, ”A geometric-wave way of evaluating acoustic fields in a waveguide,” Akust. Zh., 22, No.5, 724–728 (1976). [8] V. B. Philippov, ”A method for calculating the field of a point source in a waveguide,” Zap. Nauchn. Semin. LOMI, 99, 146–156 (1980). [9] A. Erdelyi and M. Wyman, ”The asymptotic evaluation of certain integrals,” ARMA, 14, No.3, 217–260 (1963). · Zbl 0168.37903 [10] L. B. Felsen, ”Backscattering from wide-angle and narrow-angle cones,” J. Appl. Phys., 26, 138–151, No. 2 (1955). · Zbl 0065.20301 [11] L. B. Felsen, ”Hybrid ray-mode fields in inhomogeneous waveguides and ducts,” J. Acoust. Soc. Amer., 69, 352–361 (1981). [12] D. Ludwig, ”Diffraction by a circular cavity,” J. Math. Phys., 11, No.5, 1617–1630 (1970). [13] V. S. Buldyrev and V. S. Buslaev, ”Asymptotic methods in problems of sound propagation in an ocean waveguide and their numerical realization,” Zap. Nauchn. Semin. LOMI, 117, 39–77 (1981). · Zbl 0482.76078 [14] V. S. Buslaev, ”Structure of the acoustic field near the surface of a deep sea,” Zap. Nauchn. Semin. LOMI, 117 (1981), 98–111. · Zbl 0482.76077 [15] V. S. Buldyrev and V. S. Buslaev, ”A qualitative structure of the acoustic field in the ocean,” Preprint Inst. Radiotech. Electronics AN SSSR, No. 44(416) (1984). · Zbl 0531.76089 [16] V. S. Buldyrev and V. S. Buslaev, ”Sound propagation in the ocean,” Preprint Inst. Radiotech. Electronics AN SSSR, No. 45(417) (1984). · Zbl 0531.76089 [17] A. G. Alenitsyn, V. S. Buldyrev, V. E. Grikurov, and A. A. Fedotov, ”Sound pulse simulation in the deep ocean,” J. Comp. Acoust., 3, No.1, 57–68 (1995). · Zbl 06713016 [18] L. M. Brekhovskikh, Acoustics of the Ocean [in Russian], Nauka, Moscow (1974). · Zbl 0744.76003 [19] L. M. Brekhovskikh, Waves in Layered Media [in Russian], Nauka, Moscow (1973). · Zbl 0558.73018 [20] A. V. Lazuta, S. U. Slavyanov and I. B. Tampel, ”On propagation of waves in doubly-connected waveguides with radiation,” in: Vth All-Union Symposium on the Diffraction and Propagation of Waves (Rostov-na-Donu), Leningrad (1971), pp. 125–133. [21] N. Ya. Kirpichnikova, V. B. Philippov, and N. G. Vlasyuk, ”A combined method of calculating the field in a waveguide, initiated by a point source,” Zap. Nauchn. Semin. POMI, 257, 66–74 (1999). · Zbl 0985.78018 [22] V. S. Buldyrev, ”The field of a point source in a waveguide,” Tr. Mat. Inst. Akad. Nauk SSSR, 115, 78–102 (1971). · Zbl 0237.35024
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