Kirchhoff formula

Kirchhoff integral

The formula

$$ \tag{1 } u ( x , t ) = \frac{1}{4 \pi } \int\limits _ \Omega \frac{f ( y , t - r ) }{r } d \Omega _ {y} + $$

$$ + \frac{1}{4 \pi } \int\limits _ \sigma \left [ \frac{1}{r} \frac{\partial u }{\partial n } - u \frac{\partial ( 1 / r ) }{\partial n } + \frac{1}{r} \frac{\partial u }{\partial \tau } \frac{\partial r }{\partial n } \right ] _ {\tau = t - r } d \sigma _ {y} , $$

expressing the value $ u ( x , t ) $ of the solution of the inhomogeneous wave equation

$$ \tag{2 } u _ {tt} - u _ {x _ {1} x _ {1} } - u _ {x _ {2} x _ {2} } - u _ {x _ {3} x _ {3} } = f ( x , t ) $$

at the point $ x =( x _ {1} , x _ {2} , x _ {3} ) \in \Omega $ at the instant of time $ t $ in terms of the retarded volume potential

$$ v _ {1} ( x , t ) = \ \frac{1}{4 \pi } \int\limits _ \Omega \frac{f ( y , t , r ) }{r } d \Omega _ {y} ,\ \ y = ( y _ {1} , y _ {2} , y _ {3} ) , $$

with density $ f $, and in terms of the values of the function $ u ( y , t ) $ and its first-order derivatives on the boundary $ \sigma $ of the domain $ \Omega $ at the instant of time $ \tau = t - r $. Here $ \Omega $ is a bounded domain in the three-dimensional Euclidean space with a piecewise-smooth boundary $ \sigma $, $ n $ is the outward normal to $ \sigma $ and $ r = | x - y | $ is the distance between $ x $ and $ y $.

Let

$$ v _ {1} ( x , t ) = \ \frac{1}{4 \pi } \int\limits _ \sigma \frac{1}{r} \frac{\partial \mu _ {1} ( y , t - r ) }{\partial n } d \sigma _ {y} , $$

$$ v _ {2} ( x , t ) = \frac{1}{4 \pi } \int\limits _ \sigma \frac{\partial ^ {*} }{\partial ^ {*} n } \frac{ \mu _ {2} ( y , t - r ) }{r } d \sigma _ {y} , $$

where

$$ \frac{\partial ^ {*} }{\partial ^ {*} n } \frac{\mu _ {2} ( y , t - r ) }{r } = \ \frac{1}{r} \frac{\partial r }{\partial n } \frac{\partial \mu _ {2} ( y , t - r ) }{\partial t } - \mu _ {2} ( y , t - r ) \frac{\partial ( 1 / r ) }{\partial n } . $$

The integrals $ v _ {1} ( x , t ) $ and $ v _ {2} ( x , t ) $ are called the retarded potentials of the single and the double layer.

The Kirchhoff formula (1) means that any twice continuously-differentiable solution $ u ( x , t ) $ of equation (2) can be expressed as the sum of the retarded potentials of a single layer, a double layer and a volume potential:

$$ u ( x , t ) = v _ {1} ( x , t ) + v _ {2} ( x , t ) + v _ {3} ( x , t ) . $$

In the case when $ u ( x , t ) = u ( x ) $ and $ f ( x , t ) = f ( x ) $ do not depend on $ t $, the Kirchhoff formula takes the form

$$ u ( x) = \frac{1}{4 \pi } \int\limits _ \Omega \frac{f ( y ) }{r } d \Omega _ {y} + \frac{1}{4 \pi } \int\limits _ \sigma \left [ \frac{1}{r} \frac{\partial u ( y) }{\partial n } - u ( y) \frac{\partial ( 1 / r ) }{\partial n } \right ] d \sigma _ {y} $$

and gives a solution of the Poisson equation $ \Delta u = - f( x) $.

The Kirchhoff formula is widely applied in the solution of a whole series of problems. For example, if $ \Omega $ is the ball $ | y - x | \leq t $ of radius $ t $ and centre $ x $, then formula (1) is transformed into the relation

$$ \tag{3 } u ( x , t ) = \ \frac{1}{4 \pi } \int\limits _ {r \leq t } \frac{f ( y , t - r ) }{r } \ d y + t M _ {t} [ \psi ] + \frac \partial {\partial t } t M _ {t} [ \phi ] , $$

where

$$ M _ {t} [ \phi ] = \ \frac{1}{4 \pi } \int\limits _ {| y | = 1 } \phi ( x + t y ) d s _ {y} $$

is the average value of $ \phi ( x) $ over the surface of the sphere $ | y - x | = t $,

$$ \tag{4 } \left . \phi ( x) = u \right | _ {t = 0 } ,\ \ \left . \psi ( x ) = u _ {t} \right | _ {t = 0 } . $$

If $ \phi ( x) $ and $ \psi ( x) $ are given functions in the ball $ | x | \leq R $, with continuous partial derivatives of orders three and two, respectively, and $ f ( x , t ) $ is a twice continuously-differentiable function for $ | x | < R $, $ 0 \leq t \leq R - | x | $, then the function $ u ( x , t ) $ defined by (3) is a regular solution of the Cauchy problem (4) for equation (2) when $ | x | < R $ and $ t < R - | x | $.

Formula (3) is also called Kirchhoff's formula.

The Kirchhoff formula in the form

$$ u ( x , t ) = \ t M _ {t} [ \psi ] + \frac \partial {\partial t } t M _ {t} [ \phi ] $$

for the wave equation

$$ \tag{5 } \Delta u = u _ {tt} $$

is remarkable in that the Huygens principle does follow from it: The solution (wave) $ u ( x , t ) $ of (5) at the point $ ( x , t ) $ of the space of independent variables $ x _ {1} , x _ {2} , x _ {3} , t $ is completely determined by the values of $ \phi $, $ \partial \phi / \partial n $ and $ \psi $ on the sphere $ | y - x | = t $ with centre at $ x $ and radius $ | t | $.

Consider the following equation of normal hyperbolic type:

$$ \tag{6 } \sum _ {i , j = 1 } ^ { {m } + 1 } a ^ {ij} ( x) u _ {x _ {i} y _ {j} } + \sum _ { j= } 1 ^ { m+ } 1 b ^ {j} ( x) u _ {x _ {j} } + c ( x) u = \ f ( x) $$

with sufficiently-smooth coefficients $ a ^ {ij} ( x) $, $ b ^ {j} ( x) $, $ c ( x) $, and right-hand side $ f ( x) $ in some $ ( m + 1 ) $- dimensional domain $ \Omega _ {m+} 1 $, that is, a form

$$ \sum _ {i , j = 1 } ^ { {m } + 1 } a ^ {ij} ( x) \xi _ {i} \xi _ {j} $$

that at any point $ x \in \Omega _ {m+} 1 $ can be reduced by means of a non-singular linear transformation to the form

$$ y _ {0} ^ {2} - \sum _ { i= } 1 ^ { m } y _ {i} ^ {2} . $$

The Kirchhoff formula generalizes to equation (6) in the case when the number $ m + 1 $ of independent variables $ x _ {1} \dots x _ {m+} 1 $ is even [4]. Here the essential point is the construction of the function $ \phi ^ {(} k) $ that generalizes the Newton potential $ 1/r $ to the case of equation (6). For the special case of equation (6),

$$ \tag{7 } u _ {tt} - \sum _ { i= } 1 ^ { m } u _ {x _ {i} x _ {i} } = 0 ,\ \ m \equiv 1 ( \mathop{\rm mod} 2 ), $$

the generalized Kirchhoff formula is

$$ \tag{8 } u ( x , t ) = \gamma \int\limits _ \sigma \sum _ { i= } 1 ^ { k } ( - 1 ) ^ {k} \left \{ \frac{\partial \phi ^ {(} k- i+ 1) }{\partial n } \left [ \frac{\partial ^ {i-} 1 u }{\partial t ^ {i-} 1 } \right ] \right . - $$

$$ - \left . \phi ^ {(} k- i+ 1) \left [ \frac{\partial ^ {i} u }{\partial n \partial t ^ {i-} 1 } - \frac{\partial r }{\partial n } \left [ \frac{\partial ^ {i} u }{\partial t ^ {i} } \right ] \right ] \right \} d \sigma _ {x} , $$

where $ \gamma $ is a positive number, $ \sigma $ is the piecewise-smooth boundary of an $ m $- dimensional bounded domain $ \Omega _ {m} $ containing the point $ y $ in its interior, and $ n $ is the outward normal to $ \sigma $. Further,

$$ \phi ^ {(} i) = \gamma _ {i} r ^ {-} k- i+ 1 ,\ \ \phi ^ {(} k) = r ^ {2-} m ,\ \ r = | y - x | ; $$

$$ \gamma _ {i} = \textrm{ const } ,\ i = 1 \dots k - 1 ; \ k = m- \frac{1}{2} ; $$

and $ [ \psi ] $ denotes the retarded value of $ \psi ( x , t ) $:

$$ [ \psi ( x , t ) ] = \psi ( x , t - r ) . $$

Formula (8) for equation (6) is sometimes called the Kirchhoff–Sobolev formula.

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