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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|Helmholtz equation]] and consider the problem of the scattering of a plane wave by a convex surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k0554501.png" /> on which the classical (Dirichlet) boundary condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k0554502.png" /> holds. The solution reduces to finding a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k0554503.png" /> satisfying the Helmholtz equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k0554504.png" /> subject to the indicated boundary condition and representable as the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k0554505.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k0554506.png" /> satisfies the Sommerfeld [[Radiation conditions|radiation conditions]]. The solution of the problem exists and it has the integral representation
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k0554507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k0554508.png" /></td> </tr></table>
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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|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|radiation conditions]]. The solution of the problem exists and it has the integral representation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k0554509.png" /></td> </tr></table>
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$$ \tag{1 }
 +
u ( x)  = e ^ {ik x _ {1} } -  
 +
\frac{1}{4 \pi }
 +
\int\limits _  \Sigma 
 +
\frac{\partial
 +
u ( x  ^  \prime  ) }{\partial  n _ {x  ^  \prime  } }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545010.png" /> is the derivative along the normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545011.png" />. The normal is taken outward relative to the infinite domain bounded internally by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545012.png" />. It is assumed that on the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545013.png" /> illuminated by the plane wave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545015.png" /> is approximately equal to the expression obtained by the [[Ray method|ray method]]. On the shadowed part one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545016.png" />. The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545017.png" /> obtained in this way is called the Kirchhoff approximation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545018.png" />.
+
\frac{e ^ {ik | x - x  ^  \prime  | } }{| x - x  ^  \prime  | }
 +
  d \Sigma _ {x  ^  \prime  } ,
 +
$$
  
In the illuminated region, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545019.png" /> and the [[Geometric approximation|geometric approximation]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545020.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545021.png" /> is expressed in terms of the Fresnel integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545022.png" />, and in the shadowed zone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545023.png" /> (in fact in the shadowed zone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545024.png" /> decreases considerably faster than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545025.png" />).
+
$$
 +
x  =  ( x _ {1} , x _ {2} , x _ {3} ) ,\  x  ^  \prime  = \
 +
( x _ {1}  ^  \prime  , x _ {2}  ^  \prime  , x _ {3}  ^  \prime  ) ,
 +
$$
  
The Kirchhoff method gives a formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545026.png" /> that is correct in the principal terms and remains correct as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545027.png" />. In the subsequent orders in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055450/k05545028.png" /> the Kirchhoff approximation is no longer applicable.
+
$$
 +
| x - x  ^  \prime  |  =  \sqrt {\sum _ { i= } 1 ^ { 3 }  ( x _ {i} - x _ {i}  ^  \prime  )  ^ {2} } ,
 +
$$
 +
 
 +
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|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|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hönl,  A.-W. Maue,  K. Westpfahl,  "Theorie der Beugung"  S. Flügge (ed.) , ''Handbuch der Physik'' , '''25/1''' , Springer  (1961)  pp. 218–573</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hönl,  A.-W. Maue,  K. Westpfahl,  "Theorie der Beugung"  S. Flügge (ed.) , ''Handbuch der Physik'' , '''25/1''' , Springer  (1961)  pp. 218–573</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Rubinowicz,  "Die Beugungswelle in der Kirchhoffschen Theorie der Beugung" , PWN  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Rubinowicz,  "Die Beugungswelle in der Kirchhoffschen Theorie der Beugung" , PWN  (1957)</TD></TR></table>

Revision as of 22:14, 5 June 2020


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

Comments

References

[a1] A. Rubinowicz, "Die Beugungswelle in der Kirchhoffschen Theorie der Beugung" , PWN (1957)
How to Cite This Entry:
Kirchhoff method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kirchhoff_method&oldid=15726
This article was adapted from an original article by V.M. Babich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article