Exact mode representation of sound propagation in a shallow-water wedge with a penetrable bottom

Abstract

A new solution for the acoustic wave propagating in a wedge-shaped shallow-water waveguide with a penetrable bottom (ρ1≠ρ2, c1≠c2) is presented. By using a special transformation of range function (Bessel function), the inherent nonseparable problem can be converted to a pseudoseparable one. The solution is exact and is expressed as a summation of wedge mode sets, i.e., each eigenvalue corresponds to a set of eigenfunctions. There are several advantages of the present method over the conventional coupling mode method: (1) since the cylindrical coordinates corresponding to wedge modes are used, the media are bounded in angular direction 0←θ←π; therefore, the mode spectrum of the exact solution only contains discrete modes. (2) The diffracted wave from wedge apex is included automatically. The contribution of the diffracted wave to the total field is important when kr≪1, where k is the wave number and r is the range from the wedge apex. (3) The mode coefficients are obtained through recursive formulas instead of solving M linear equations simultaneously, where M is the number of modes. When c1=c2, the problem becomes separable and for each eigenvalue, only the primary eigenfunction corresponding to a conventional wedge mode remains nonzero. Hence, the solution smoothly reduces to that for an isovelocity/density contrast wedge [D. Chu, J. Acoust. Soc. Am. 87, 2442–2450 (1990)]. Numerical computations are presented to show the effect of mode coupling in terms of different geometrical and physical parameters.