Tailored Green’s Functions – The Rigid Half-Plane
Exact tailored Green’s functions which remain tractable for analytical modeling are very few in mathematical wave theory. Some of them can be generated from the free-space Green’s function by the method of images, taking advantage of the fact that reflecting plates can be removed provided that the symmetric images of the primary sources are introduced (the principle has already been used in Fig. 4). The half-space bounded by an infinite, either soft or rigid wall, and the quarter-space defined by two perpendicular planes, can be treated this way. Similar is the case of the channel limited by two parallel planes, if the infinite set of sources corresponding to the successive reflections is considered. However the channel is better considered as a waveguide, and the Green’s function expressed as a combination of the so-called acoustic modes of propagation (see for instance Goldstein (1976) for sound propagation in ducts). Other tailored Green’s functions useful when formulating open-air radiation problems are also available for the space limited by a wedge of arbitrary angle, the halfplane being the special case of wedge with external angle 2 n (Macdonald (1915)). Quite obviously, deriving tailored Green’s functions for more complicated shapes can be as difficult as solving the full problem and is often accessible only through numerical implementations of the wave equation or of the Helmholtz equation. Therefore approximate Green’s functions are an interesting alternative when they can be defined, for instance by removing the observer at very large distances. A class of such approximations, not discussed here, is provided by Howe’s so-called compact Green’s functions (Howe (2003)), often addressing sources very close to compact solid bodies and observers in the acoustic far field. Since the tailored Green’s functions rely on source and/or observer distances scaled by the acoustic wavelength, they are more specifically associated with the Helmholtz equation and the frequency-domain approach.