Integral equation for the acoustic wavefield

As in the electromagnetic case, the differential equation (14.13) can be transformed to an integral equation for the scattered field. Certainly, let us recall the integral 

Substituting V p and V G from equations (14.10) and (14.15) into the last equation, we find 

We have demonstrated in a previous chapter that, according to the Sommerfeld radiation conditions (14.4), if the radius R is expanded without limit, the surface integral over Or tends to zero. As a result, we arrive at expression (14.14). 

Using the same technique, we can derive a similar integral representation for the scattered field as well  

Substituting the expression for the anomalous source (14.12) into (14.18), we finally obtain the well-known representation of the scattered field as an integral over the anomalous source in inhomogeneous domain D (Bleistein, 1984)  

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