 Rayleigh integral for baffled acoustic radiation

An aperture in a rigid plane with prescribed harmonic normal displacement $f(x^{\prime })e^{i\omega t}$ radiates into a half-space. Superposing acoustic monopoles, each doubled by its same-sign rigid-wall image, gives the stated integral, where $R$ is the distance from aperture element to observer. In the acoustic far field, expanding $R$ to first order across the aperture identifies the angular radiation amplitude with the aperture's Fourier transform.