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 $\Sigma$ on which the classical (Dirichlet) boundary condition $u \mid _ \Sigma = 0$ holds. The solution reduces to finding a function $u$ satisfying the Helmholtz equation $( \Delta + k ^ {2} ) u = 0$ subject to the indicated boundary condition and representable as the sum $u = e ^ {ikx _ {1} } + U$, where $U$ satisfies the Sommerfeld radiation conditions. The solution of the problem exists and it has the integral representation

$$\tag{1 } u ( x) = e ^ {ik x _ {1} } - \frac{1}{4 \pi } \int\limits _ \Sigma \frac{\partial u ( x ^ \prime ) }{\partial n _ {x ^ \prime } } \frac{e ^ {ik | x - x ^ \prime | } }{| x - x ^ \prime | } d \Sigma _ {x ^ \prime } ,$$

$$x = ( x _ {1} , x _ {2} , x _ {3} ) ,\ x ^ \prime = \ ( x _ {1} ^ \prime , x _ {2} ^ \prime , x _ {3} ^ \prime ) ,$$

$$| x - x ^ \prime | = \sqrt {\sum _{i=1}^ { 3 } ( x _ {i} - x _ {i} ^ \prime ) ^ {2} } ,$$

where $\partial / \partial n _ {x ^ \prime }$ is the derivative along the normal to $\Sigma$. The normal is taken outward relative to the infinite domain bounded internally by $\Sigma$. It is assumed that on the part of $\Sigma$ illuminated by the plane wave $e ^ {ikx _ {1} }$, $\partial u / \partial n _ {x ^ \prime }$ is approximately equal to the expression obtained by the ray method. On the shadowed part one sets $\partial u ( x ^ \prime ) / \partial n _ {x ^ \prime } = 0$. The expression $u _ {K}$ obtained in this way is called the Kirchhoff approximation for $u$.

In the illuminated region, $u _ {K}$ and the geometric approximation for $u$ 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 $u _ {K}$ is expressed in terms of the Fresnel integral $\int _ {0} ^ \infty e ^ {i \alpha ^ {2} } d \alpha$, and in the shadowed zone $u _ {K} = O ( 1/ k )$( in fact in the shadowed zone $u$ decreases considerably faster than $1 / k$).

The Kirchhoff method gives a formula for $u$ that is correct in the principal terms and remains correct as $| x | \rightarrow \infty$. In the subsequent orders in $k$ the Kirchhoff approximation is no longer applicable.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
[a1]	A. Rubinowicz, "Die Beugungswelle in der Kirchhoffschen Theorie der Beugung" , PWN (1957)