Application of a variational principle to acoustic radiation from a vibrating finite cylinder

Abstract

A variational principle derived from the Kirchhoff‐Helmholtz integral corollary is applied to a rigid cylinder; previous applications have been to acoustic radiation and diffraction from an infinitely thin disk [J. H. Ginsberg et al., J. Acoust. Soc. Am. Suppl. 1 77, S60 (1985)]. For a finite length circular cylinder, additional complications are encountered because of the need to consider the cylindrical sides as well as ends. The basis functions for a Rayleigh‐Ritz solution have different analytical forms on the sides and ends, although they must be continuous at the edges. A larger number of functions are required to get a good representation of the surface pressure. Calculation of the resulting matrix elements for the determination of the expansion coefficients is nontrivial because of the logrithmic singularities presented by the Green's function integrals. The surface pressure is expected to be finite, but the tangential derivative of the surface pressure is a priori known to have a s−2/3 singularity at the edges separating the side wall and ends. It is not necessary that trial functions exhibit the latter behavior, but if they are so chosen, a good convengence is expected for a smaller number of basis functions. A numerical method based on splitting each integrand into a simple singularity term plus a nonsimple but bounded term is described. Numerical results are compared with those obtained by an alternative technique developed by P. H. Rogers [NRL Report 7240 (19 June 1972)].