Two-dimensional acoustical diffraction by a penetrable wedge

A problem that has consistently defied rigorous analysis is the construction of the time-periodic Green’s function for an acoustic line source in the presence of a penetrable wedge with distinct wave speed. This paper begins by noting that in the ‘‘soft’’ limit (density ratio becoming zero) the interior field must become null and hence it is sensible to express the required Green’s function as a series in powers of the density ratio. Applications of Green’s theorem then reduce the problem to a sequence of integral equations for the determination of the coefficients in a similar expansion of the normal velocity at the wedge boundary. The exterior ‘‘hard’’ Green’s function now appears as the kernel while the interior hard Green’s function appears in the forcing terms to take account of penetration into the wedge. After observing that the velocity coefficients can be calculated recursively in terms of the solution of a single integral equation of the first kind, the latter is itself transformed to an equation of the second kind by using a known technique for Hankel function kernels. Some geometrical acoustics features of the solution are described and a method for subsequently determining the far field scattering pattern is given.