A Monopole Point Source Above a Bulk Reacting Plane, Exact Forms

→ See also Mechel, Vol. I, Ch. 13 (1989)

A monopole point source is placed at a point Q at height h above an absorbent plane; a field point is at P. The plane may be bulk reacting. See → Sect.  D.17 for a locally reacting plane. See Mechel  (1989) for references to the extensive literature about this problem. As an exception, the time factor in this section is $e^{{-i\,\omega t}}$ in order to facilitate the comparison with the literature, where this sign convention mostly is used.

The free field of the point source is $p_{Q}(r_{1})=\displaystyle P_{0}\frac{e^{{i\, k_{0}r_{1}}}}{k_{0}r_{1}}\;,$ (1)

the field p above the absorber is $\displaystyle\frac{p}{P_{0}}=\displaystyle\frac{e^{{i\, k_{0}r_{1}}}}{k_{0}r_{1}}+\frac{p_{r}}{P_{0}}\;,$ (2)

with $r_{{1}}=dist(Q,P)$ and $p_{{r}}$ the reflected field. The task is to find $p_{{r}}$.

An exact integral expression for $p_{{r}}$ is

$$\frac{p_{r}}{P_{0}}=i\int\limits...