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minus bold r 0 right-parenthesis"/> (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

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