Kirchhoff integral theorem


Kirchhoff's integral theorem uses Green's identities to derive the solution to the homogeneous wave equation at an arbitrary point P in terms of the values of the solution of the wave equation and its first-order derivative at all points on an arbitrary surface that encloses P.

Equation

Monochromatic waves

The integral has the following form for a monochromatic wave:
where the integration is performed over an arbitrary closed surface S, s is the distance from the surface element to the point r, and ∂/∂n denotes differentiation along the surface normal. Note that in this equation the normal points to the inner of the enclosed volume; if the more usual outer-pointing normal is used, the integral will have the opposite sign.

Non-monochromatic waves

A more general form can be derived for non-monochromatic waves. The complex amplitude of the wave can be represented by a Fourier integral of the form
where, by Fourier inversion, we have
The integral theorem is applied to each Fourier component, and the following expression is obtained:
where the square brackets on V terms denote retarded values, i.e. the values at time t − s/c.
Kirchhoff showed that the above equation can be approximated in many cases to a simpler form, known as the Kirchhoff, or Fresnel–Kirchhoff diffraction formula, which is equivalent to the Huygens–Fresnel equation, but provides a formula for the inclination factor, which is not defined in the latter. The diffraction integral can be applied to a wide range of problems in optics.