โ Scribed by David H. Berman
No coin nor oath required. For personal study only.
Using energy conservation and causality considerations, the completeness of scattering states is established for plane waves impinging on an irregular interface. Provided certain limiting operations commute with differentiation, it is shown that surface waves need not be explicitly included in the Weyl representation of the Green's function in the presence of a rough interface. Rather, surface waves are implicitly included through the poles of the scattering amplitudes. This result was used implicitly in a recently developed scheme to treat scattering in a duct using half-space scattering amplitudes (Berman, J. Acoust. Sot. Am. 92 (1992), 309-314).
๐ SIMILAR VOLUMES
From this one finds for g+(z) = q(z)/p(z) This function can be directly checked to be a solution of (I 1), so that in the case (21) the set (1) is not complete.
This result could also be obtained by using in the work of the previous section other solutions of {25) than ( ). The special solution ( ) is uniquely characterized by the property that S0{x)/0\_(x) belongs to ~\_. Indeed, a solution ho\_(x) of the homogeneous equation cannot have this same property