Asymptotic approximation of waves due to a dipole on a two‐layer medium

The TM wave due to a horizontal magnetic dipole is first expanded in terms of a series of integrals, each of which is interpreted as due to an image source. Each of the image source integrals possesses a branch point and a high-order pole which can be close to the saddle point. To facilitate uniform asymptotic approximation, the branch point is first removed by a transformation, which results in an integral with a high-order pole near three saddle points. The uniform asymptotic approximation of such an integral is obtained through the use of the generalized Weber's function. The result agrees with numerical integration much better than the geometrical optics approximations (GOA). For the case of TE waves, the pole is far away, and our analysis reduces to the three-saddle-point analysis where the behavior of the wave can be approximated by parabolic cylinder functions. We show that in such a case our result agrees better with the experiment than GOA. The uniform asymptotic approximation of the waves extends the domain of validity of the image source representation, which is extremely useful when the layer is thick.