Acoustics: Sound Fields, Transducers and Vibration, 2nd Edition

13.2. The Rayleigh integrals and Green's function

In this section, monopole and dipole boundary integrals are derived in an intuitive way based on the Huygens–Fresnel principle whereby monopole and dipole point sources are summed over surfaces. A more mathematically rigorous treatment follows in Section 13.3. We have treated all problems so far in this text as boundary value problems based on solutions to the following homogeneous Helmholtz wave equation in an arbitrary coordinate system:

( ∇ 2 + k 2 ) p ˜ ( r ) = 0 ,

(13.1)

where, for example, in rectangular coordinates (r)   =   (x,y,z) and the Laplace operator ...