A Monopole Point Source Above a Locally Reacting Plane, Approximations

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

See → Sect.  D.17 for exact integral formulations of the solution, and see Mechel  (1989) for a discussion of the approximations and their precision.

A monopole point source is placed at a point Q at height h above an absorbent plane; a field point is at P with height z above the plane and horizontal distance r from Q. $Q^{\prime}$ is the mirror-reflected point to Q. The plane is locally reacting with a normalised admittance $G=1/Z$.

See Mechel  (1989) for a detailed discussion of the approximations. As an exception, the time factor in this section is $e^{{-i\omega t}}$ in order to facilitate comparison with the literature, where this sign convention mostly is used. Radii used are $r_{{1}}=dist(Q,P)$ and $r_{{2}}=dist(Q^{\prime},P)$, and the angle $\Theta _{0}=\angle\left({({Q}^{{\prime}},Q),({Q}^{{\prime}},P)}\right)$.

The start equation for a first approximation to the reflected field p ${}_{{r}}$ is

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