In this paper a rigorous mathematical proof is given of the so-called limiting amplitude principle for reflecting bodies. This principle states that every solution of the wave equation with a harmonic forcing term,
uniformly on bounded sets as t tends to infinity. Here V satisfies the reduced wave equation
vanishes on the body and satisfies the Sommerfeld radiation condition at infinity. In the precise formulation of the result some conditions are imposed of which the most important is that the body is star-shaped.
The proof is based on new estimates for solutions of the homogeneous wave equation which vanish on the body. These estimates are derived from so-called energy identities, i.e. identities which express the space-time integral in the exterior of a reflecting body tends to the steady state solution in terms of space and space-time integrals of quadratic forms (Friedrichs' a, b, c-method). Here N denotes a first order operator whose direction of differentiation points into the interior of the characteristic cone of the wave operator. It is of paramount importance to choose this auxiliary operator appropriately.
From the resulting energy identity we can estimate the rate of decay of the energy contained in a given finite region' and from this we derive pointwise estimates of the decay of the solution by using Sobolev's inequality and the Kirchoff formula for solutions of the wave equation. The operator N used in this paper differs from the one used in our previous paper [l]; the estimates derived there are weaker than the present results.
* This paper represents result obtained under the sponsorship of the U. S. Army Research Office (Durham) DA-ARO(D)-31-124.G156. Reproduction in whole or in part permitted for any purpose of the United States Government.
i It has been shown by Lax and Phillips [4] using entirely different methods that for any smooth body the solution of the mixed initial value problem decays.