Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 75

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

(2.138)

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

(2.139)

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

(2.140)

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).

(2.141)