language-icon Old Web
English
Sign In

Kirchhoff integral theorem

Kirchhoff's integral theorem (sometimes referred to as the Fresnel–Kirchhoff 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. Kirchhoff's integral theorem (sometimes referred to as the Fresnel–Kirchhoff 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. The integral has the following form for a monochromatic wave: where the integration is performed over an arbitrary closed surface S (enclosing r), s is the distance from the surface element to the point r, and ∂/∂n denotes differentiation along the surface normal (a normal derivative). 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. 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 (above) is applied to each Fourier component U ω {displaystyle U_{omega }} , 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.

[ "Electric-field integral equation", "Summation equation", "Integral equation", "Partial differential equation" ]
Parent Topic
Child Topic
    No Parent Topic