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.

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