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

→ See also Mechel, Vol. I, Ch. 13 (1989); Ochmann  (2004)

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 is locally reacting with a normalised admittance $G=1/Z$. See → Sect.  D.16 for a bulk reacting plane; some of the forms for the field above the absorbent plane in that section can be used also for a locally reacting plane, if such forms apply the reflection factor R of a plane wave at the 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 $\displaystyle p_{Q}(r_{1})=\displaystyle P_{0}\frac{e^{{i\, k_{0}r_{1}}}}{k_{0}r_{1}}\;,$ (1)

the field p above the absorber is \(\displaystyle\frac{p}{P_{0}}=\displaystyle\frac{e^{{i\,...