Kirchhoff–Helmholtz integral

The Kirchhoff–Helmholtz integral combines the Helmholtz equation with the Kirchhoff integral theorem^[1] to produce a method applicable to acoustics,^[2] seismology^[3] and other disciplines involving wave propagation.

It states that the sound pressure is completely determined within a volume free of sources, if sound pressure and velocity are determined in all points on its surface.

${\displaystyle {\boldsymbol {P}}(w,z)=\iint _{dA}\left(G(w,z\vert z'){\frac {\partial }{\partial n}}P(w,z')-P(w,z'){\frac {\partial }{\partial n}}G(w,z\vert z')\right)dz'}$