Kirchhoff formula for an unbounded domain

Let wavefield F(r, t) satisfy the homogeneous wave equation in that domain  

The corresponding field p(r, cu), obtained as a result of a Fourier transform from the time to the frequency domain, satisfies the homogeneous Helmholtz equation  

We also assume that the frequency domain field p(r, ui) decays at infinity, 

We will show that under these assumptions the field P(r, t) may be represented inside the domain V by the Kirchhoff formulae (13.112) or (13.168). 

Since domain Vr is bounded, the field P(r, t) can be represented by the Kirch-hoff formula (13.170)  

---