A simple approximation of transmitted wavefront shape from point sources above lossy half spaces

Many subsurface sensing problems require knowledge of the shape of a wavefront in a half-space originating from a point source above the half-space. Impulse ground penetrating radar and air-coupled acoustic underground sensing systems usually have their emitters in air, transmitting waves through a quasi-planar interface into soil or water. The waves radiated by the emitter can be approximated as circular, but once they enter the half-space, they propagate with wavefront shape that resembles a hyperbola. The objective of this paper is to derive the exact shape and show that it is well approximated by a hyperbola with specific parameters depending on the dielectric constant of the half-space epsi´(=n²) the height of the source h, and the propagation time from source to wavefront multiplied by the wave velocity in the half space p. In addition, a correction formula is provided to reduce the error between the approximate hyperbola and the exact shape