Bragg-modulated hybrid ray-mode algorithm for sound scattering from a periodically ribbed submerged elastic cylindrical shell: Theory and asymptotics

Abstract

A wave-based hybrid ray-mode algorithm has recently been developed to parametrize mid- and high-frequency sound scattering from submerged empty and simply loaded cylindrical shells [Felsen et al., J. Acoust. Soc. Am. 87, 554–569 (1990)]. The present study extends the hybrid ray-mode scheme to acoustic scattering from a submerged infinite cylindrical elastic shell with annular ribs attached internally at periodic intervals. Assuming thin shell dynamics, the problem is solved by adapting to the nonperiodic azimuthally unbounded domain (−∞<φ<∞) the azimuthally periodic harmonic series procedure used by Burroughs [J. Acoust. Soc. Am. 75, 715–722 (1984)] for this problem; the unbounded azimuthal (φ) domain with its continuous wave-number spectrum is the essential starting point for the wave-based ray-acoustic analysis. The solution in the spectral domain is separated into the previously derived contributions for the nonloaded shell and into contributions from the infinite array of periodic ring forcings on the shell surface that account for the internal loads. Spectral synthesis furnishes the formal integral representation that is then manipulated by integration path deformations into a form amenable to reduction by saddle point asymptotics and residue calculus so as to yield the desired hybrid ray-mode combination. The final algorithm for the scattered field contains (a) specular and Bragg-modulated diffracted ray fields; (b) new unmodulated and Bragg-modulated helically traveling shell-guided trapped, creeping and leaky modes, which are phase-matched to the fluid, in all wave species (compressional, flexural, shear) that the loaded thin shell geometry can support; (c) helically resonant shell-guided modes, formed by phase coherent circumnavigations of the loaded cylinder by the traveling modes in (b). The dispersion equations for the various Bragg-modulated wave types and species are determined from the internal impedance functions for the periodically distributed loadings.