An exact Laplace transform formulation for a point source

An exact analysis is given for a point source in air above a ground surface. By representing the plane-wave reflection coefficient as the Laplace transform of an image source distribution, a well-behaved image integral, instead of the usual Sommerfeld integral, is obtained. The approach is valid for both locally and extended reacting surfaces. For a locally reacting ground surface, the image integral is an especially simple, rapidly convergent integral. The integral for local reaction is investigated analytically for a number of limiting cases. The resulting analytic solutions are compared with analytic solutions obtained from more standard approaches. Finally, the image integral for local reaction is analyzed numerically, and an upper limit on the numerical integration is given. It is shown that with realistic values of ground impedance, the prescribed integration limit allows the image integral to be easily and accurately computed numerically.