On the acoustic field in a Pekeris waveguide with attenuation in the bottom half-space

The acoustic field in a Pekeris channel with an attenuating basement is critically examined, based on contour integrations of the wave number integrals for the two domains. In both regions, the field consists of a finite sum of proper (square integrable) normal modes plus a branch line integral around a hyperbolic cut. For low bottom attenuation, only "trapped" modes exist but as the loss increases additional "dissipation" modes contribute to the mode sum. A Newton-Raphson iterative procedure is introduced for finding the complex eigenvalues of the modes and a new expression is derived which shows that the total number of proper (trapped plus dissipation) modes supported by the waveguide increases essentially linearly with rising bottom attenuation. Approximations for the complex shape functions of the modes in the water column and the basement are developed and compared with the exact shape functions determined from the Newton-Raphson procedure. An expression derived for the modal attenuation coefficient scales in proportion to the square of the mode number and inversely with the square of the frequency. Stationary-phase approximations for the branch line integrals for both domains are developed, which serve to illustrate the asymptotic range dependence of the lateral wave but otherwise have little utility.