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 infinite ground plane is solved analytically for both the TM and TE polarizations. A quasi-static solution for the inner field based upon a transformation to bipolar coordinates exploits the failure of the narrow trough to react to the detailed wave nature of the incident field and forms the starting point for the method of matched asymptotic expansions. The distant behavior of the inner field must agree with the near behavior of the outer field, which is a radiative solution of the Helmholtz equation. In addition to yielding several analytic terms of the solution in low-order powers and the logarithm of the trough wave size kα the matching process provides an account of the interplay between all of the physical parameters