S"/> (2.136)
some volume integrals can be transferred into surface integrals and we get finally
The first term on the right-hand side is the volume integral over all sources fq(r) in the volume. So given a known source distribution we can calculate the according sound field. The two terms in the surface integral take care of the boundary condition. The pressure gradient in the first can be converted into the normal velocity using (2.35). The second surface integral allows establishing the correct surface impedance. Equation (2.137) is called the constant frequency version of the .
2.7.2 Rayleigh integral
The Rayleigh integral is a special solution of the Kirchhoff-Helmholtz integral applied to flat and infinite surfaces. We assume a configuration as shown in Figure 2.10. The integration volume is the right half space for z>0 closed by a half sphere of infinite radius. The Green’s function of any source at r0=(x0,y0,z0) with z0>0 is as defined in equation (2.127). The rigid surface acts as a reflector. Thus, there is a mirror source located at r0′=(x0,y0,−z0). This source is not in volume V, and the added wave field is therefore considered as a homogeneous solution in the volume. Hence, we get for the generalized Green’s function
Figure 2.10 Half space in front of a rigid wall. Source: Alexander Peiffer.
We enter this version of the Green’s function in Equation (2.137) and we get
We assume a source-free half space so fq(r)=0, and due to the mirror source symmetry ∂G(r0,r)∂z=0 is also true. By clever selection of the Green’s function we fulfilled the boundary condition automatically. For the surface integral the contributions from the half sphere with infinite radius are supposed to be zero. From Equation (2.35) the first expression can be converted into an expression for the surface velocity vz. Performing the limit process z0→0 we get
and with this Green’s function we can derive the Rayleigh integral that allows the calculation of infinite half space sound fields excited by a rigid vibrating plane with arbitrary velocity distribution vz(x0,y0).
2.7.3 Piston in a Wall
A cylindrical loudspeaker in a wall can be modelled by a piston of radius R vibrating with velocity vz located in a rigid wall. For convenience the surface integral will be expressed in cylindrical coordinates r0 and φ0. The receiver coordinates are given as spherical coordinates r and ϑ(Figure 2.11). Without loss of generality the azimuthal angle φ is set to zero.
Figure 2.11 Coordinate definitions for the piston in the wall. Source: Alexander Peiffer.
In the far field approximation we assume l≈r and get
So, the approximate result is
