Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 74

This section presents the concept of the Green’s function that uses a formalism to calculate the sound field for arbitrary source and boundary configurations as shown for example by Morse and Ingard (1968).

The Green’s function is defined as the solution of the following inhomogeneous wave equation.

(2.124)

The inhomogeneous part is the delta function which allows for this elegant derivation of the Kirchhoff integral. The delta function is introduced in the appendix A.1.3 in the time domain. However, it can also be applied in space. The multidimensional delta function is simply the product of three Dirac delta functions in space

(2.125)

The sifting properties and the value of the integration is defined by volume integral

(2.126)

The solution of Equation (2.124) is the point source (2.91)³.

(2.127)

In order to achieve a common formulation we add an arbitrary solution χ of the homogeneous wave equation

(2.128)

to the Green’s function to get the generalized Green’s function

(2.129)

The purpose of the additional homogeneous solution is to create freedom to fulfill boundary conditions that do not occur in the free sound field. The task is to find the solution for the inhomogeneous wave equation

(2.130)

The generalized Green’s function must be a solution of the following equation for the special case with r,r0∈V and the boundary ∂V as shown in Figure 2.9.

(2.131)

Figure 2.9 Solution volume and boundaries. Source: Alexander Peiffer.

In order to receive a global solution we perform the operation

(2.132)

This leads to

(2.133)

Exchanging r and r0 and integrating r0 over the volume V gives

(2.134)

The last term on the RHS follows from the sifting property of the delta function

(2.135)

With Green’s law of vector analysis

(2.136)

some volume integrals can be transferred into surface integrals and we get finally

(2.137)

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 .