Singular perturbation methods in acoustics: diffraction by a plate of finite thickness

Abstract

Singular perturbation methods are used to determine the field diffracted by a semi-infinite rigid plate of thickness $2a$ under irradiation by a plane acoustic wave at wavenumber $k$. Six terms of both an outer and an inner expansion in the small parameter $\epsilon $ = $ka$ are calculated in closed form, yielding simple results for the far-field directivity pattern. The outer series is determined by certain eigensolutions, and by a sequence of straightforward Wiener-Hopf problems, while the inner terms are all obtained from a simple conformal mapping. Previous discussions of this problem (e.g. Jones 1953) involve the formulation of a modified Wiener-Hopf equation, and reduce the problem to that of inverting an infinite matrix with elements dependent upon $\epsilon $. Jones has given a numerical inversion of the truncated $4\times 4$ matrix in the limit $\epsilon $ $\rightarrow $ $0$. Here we obtain exact expressions A$_{2n+1}$ = $\frac{1}{2(2n+1)}$ {J$_{n}$($n+\frac{1}{2}$) - J$_{n+1}$($n+$$\frac{1}{2}$)} for Jones's variables, and prove that they satisfy his infinite system of linear equations. It is also shown that to O($\epsilon $$^{2}$ ln$^{2}$ $\epsilon $) the plate may be replaced by a duct longer than the plate by an amount $L$ = ($a$/$\pi $) $ln2$, in agreement with Jones's numerical value of $L=0.22a$, together with the monopole field necessary to annul the effect of plane wave propagation down the duct. The principle used here to match the inner and outer asymptotic expansions is a slightly, though significantly, modified version of the one used extensively by Van Dyke (1964). In the present problem Van Dyke's matching principle appears to hold, in that matching can be formally accomplished, but leads to erroneous results violating the reciprocal theorem. Accordingly, an appendix here gives a discussion and justification, in elementary terms, of the proposed modified asymptotic matching principle.