Kirchhoff method
A method for approximately solving problems in the theory of the diffraction of short waves; proposed by G.R. Kirchhoff. In its simplest version Kirchhoff's method amounts to the following: Let a wave process be described by the Helmholtz equation and consider the problem of the scattering of a plane wave by a convex surface on which the classical (Dirichlet) boundary condition
holds. The solution reduces to finding a function
satisfying the Helmholtz equation
subject to the indicated boundary condition and representable as the sum
, where
satisfies the Sommerfeld radiation conditions. The solution of the problem exists and it has the integral representation
![]() | (1) |
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where is the derivative along the normal to
. The normal is taken outward relative to the infinite domain bounded internally by
. It is assumed that on the part of
illuminated by the plane wave
,
is approximately equal to the expression obtained by the ray method. On the shadowed part one sets
. The expression
obtained in this way is called the Kirchhoff approximation for
.
In the illuminated region, and the geometric approximation for
are the same in their principal terms. In a neighbourhood of the boundary between the illuminated and shadowed zones, the principal term of the asymptotic expansion of
is expressed in terms of the Fresnel integral
, and in the shadowed zone
(in fact in the shadowed zone
decreases considerably faster than
).
The Kirchhoff method gives a formula for that is correct in the principal terms and remains correct as
. In the subsequent orders in
the Kirchhoff approximation is no longer applicable.
References
[1] | H. Hönl, A.-W. Maue, K. Westpfahl, "Theorie der Beugung" S. Flügge (ed.) , Handbuch der Physik , 25/1 , Springer (1961) pp. 218–573 |
Comments
References
[a1] | A. Rubinowicz, "Die Beugungswelle in der Kirchhoffschen Theorie der Beugung" , PWN (1957) |
Kirchhoff method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kirchhoff_method&oldid=15726