Matched asymptotic expansion for the low-frequency scattering by a semi-circular trough in a ground plane

Plane wave scattering by an electrically small circular trough cut in an in finite ground plane is solved analytically for both the TM and TE polarizat ions. A quasi-static solution for the inner field based upon a transformati on to bipolar coordinates exploits the failure of the narrow trough to reac t to the detailed wave nature of the incident held and forms the starting p oint for the method of matched asymptotic expansions. The distant behavior of the inner held must agree with the near behavior of the outer field, whi ch is a radiative solution of the Helmholtz equation. In addition to yieldi ng several analytic terms of the solution in low-order powers and the logar ithm of the trough wave size ka the matching process provides an account of the interplay between all of the physical parameters.