All-frequency normal-mode solution of the three-dimensional acoustic scattering from a vertical cylinder in a plane-horizontal waveguide

The three-dimensional acoustic scattering from a vertical, impenetrable cylinder in a waveguide is studied. The analytical solution of the problem, for a Dirichlet or a Neumann boundary condition on the scatterer, has been derived recently by Athanassoulis and Prospathopoulos [J. Acoust. Soc. Am. 100, 206–218 (1996)] in the form of a double-infinite normal-mode series, representing the total acoustic field. In order to extend the applicability of this solution to higher frequencies, the total field is decomposed into the incident and the scattered parts. A series expansion for the scattered field is obtained, and the critical parameter controlling its azimuthal convergence is shown to be the nondimensional wave number ka based on the radius a of the cylinder. The general term of the series starts to decay exponentially immediately after the azimuthal index has exceeded the critical value ka, a fact justifying the introduction of the concept of azimuthal-evanescent modes. By exploiting the above decomposition, the direct numerical summation of the normal-mode series becomes feasible up to ka≈1000. For calculations at even higher frequencies (ka→∞), asymptotic expressions are derived by using appropriate integral representations of Bessel and Hankel functions, in conjunction with the method of stationary phase. The asymptotic analysis shows that the scattered field is obtained as a superposition of 2-D point sources lying on the boundary of the vertical cylinder, with appropriate amplitudes and phases. Excellent agreement between asymptotic and direct summation numerical results has been demonstrated, at moderate frequencies, where both representations are expected to be valid.