Asymptotic analysis for the diffraction of sound by closely spaced and lightly loaded semi-infinite flexible surfaces

A parallel pair of semi-infinite flexible surfaces, with equilibrium positi ons x > 0, y = 0 and x > 0, y = d, are fixed to a supporting structure at x < 0, 0 < y < d. Compressible fluid with wave speed c occupies the three re gions (i) y < 0, (ii) 0 < y < d, x > 0 and (iii) y > d outside and between the wavebearing surfaces, and a time-periodic sound wave plane of velocity potential Re{exp(ikx cos theta + iky sin theta - i omega t)} is incident up on the system. The induced vibrations depend on the type of flexible surfac e, on the details of their attachment to the supporting structure and on th e conditions imposed upon the end face at x = 0, 0 < y < d. Consideration i s given here to cases of either membranes or elastic plates, with either ac oustically soft or acoustically hard conditions at the end wall. An exact f ormulation is possible, in principle, for any values of the various paramet ers of the problem, leading to infinite systems of algebraic equations that have to be solved numerically in particular cases. The present alternative approach is to develop asymptotic solutions for the limit, of particular p hysical interest, in which the fluid loading is small and the spacing param eter d is small compared with the acoustic wavelength. Results are given, f or several different cases, in the form of integrals and as explicit asympt otic formulae which agree well with numerical evaluations of the integrals.