Sommerfeld half-space problem revisited: Short-wave asymptotic solutions

TL;DR

This paper revisits the Sommerfeld half-space problem, providing exact solutions and short-wave asymptotics using advanced integral techniques, and offers new insights into wave behavior, surface waves, and energy conservation.

Contribution

It introduces a method combining Sommerfeld integrals and saddle point approximation to derive uniform asymptotic solutions for the half-space problem.

Findings

Exact solutions in Sommerfeld integral form

Simple asymptotic expressions for space waves

Detailed analysis of surface wave existence

Abstract

A method for solving the half-space Sommerfeld problem is proposed, which allows us to obtain exact solutions in the form of Sommerfeld integrals, as well as their short-wave asymptotics. The first carried out by reducing the Sommerfeld problem to solving a system of equations to surface current densities on an interface of media in the Fourier transform domain. The second is provided by the modified saddle point method using an etalon integral. Careful attention pays the integration technique of the Sommerfeld integrals. The uniformly regular for any observation angles expressions of all types of waves, such as space, surface, and lateral, are obtained. The simple asymptotic expressions for the reflected and transmitted space waves are found via a source field, bearing in mind the Fresnel coefficients. The asymptotic expressions for the space waves checked by the law of conservation of energy flux. The issue of the existence of surface waves analyzed in detail. The analysis and physical interpretation of the obtained expressions, as well as the applicability conditions, are carried out. The Sommerfeld integrals evaluated in closed form.