On the short-wave asymptotic theory of the wave equation (∇² + k²)ø = 0

Abstract

The theory of time-periodic wave problems falls into two parts. On the one hand there is the rigorous formulation in terms of differential wave equations, on the other there are approximate theories like geometrical optics. It should be possible, in principle, to deduce the latter from the former by a logical process; but this has been done only for a few simple configurations, e.g. the circle. A possible approach to the solution of the general problem is suggested here, and is applied to a typical two-dimensional acoustical example. An arbitrary closed convex curve (satisfying certain regularity conditions) is emitting short sound waves towards infinity, the normal velocity V(s) exp (− iωt) is prescribed on the curve as a function of the arc-length s, and the potential is to be found, first on the curve and then at any point in the sound field. (Only the first part of the problem is treated in detail.) The potential ø(s) exp (− iωt) on the curve satisfies all the integral equations

where G(s, s′) is any Green's function of the problem, and V(a) is prescribed. All the equations corresponding to different Green's functions have the same solution. An asymptotic and convergent short-wave solution can be found by iteration if G can be chosen explicitly so that the integral equation has a small kernel for high frequencies. At any point of the curve draw the local circle of curvature; then the explicit known solution for a source on this circle is (with slight modifications) a possible Green's function, and the equation formed with it has a small kernel and can be solved rigorously by iteration. If V(a) is independent of the frequency, the leading term in the resulting asymptotic expansion is

where c is the velocity of sound and 2πk⁻¹ is the (short) wavelength corresponding to the frequency ω/2π. If V(a) varies rapidly, as in diffraction theory, the iterative solution still gives a convergent asymptotic expansion, but the first approximation is then practically useless in the shadow region. Diffraction problems are not treated in the present paper.

The present work appears to be the first practical and rigorous solution of a short-wave problem in optics or acoustics when a solution in closed form is not available. It is suggested that the technique (suitably combined with formal expansions) may be applicable to a wider class of radiation and diffraction problems.