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.
References
| [1] | A.V. Bitsadze, "Equations of mathematical physics" , MIR (1980) (Translated from Russian) |
| [2] | V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) |
| [3] | H. Bateman, "Partial differential equations of mathematical physics" , Dover (1944) |
| [4] | M. Mathisson, "Eine neue Lösungsmethode für Differentialgleichungen von normalem hyperbolischem Typus" Math. Ann. , 107 (1932) pp. 400–419 |
| [5] | M. Mathisson, "Le problème de M. Hadamard relatif à la diffusion des ondes" Acta Math. , 71 : 3–4 (1939) pp. 249–282 Template:ZBL |
| [6] | S.G. Mikhlin, "Linear partial differential equations" , Moscow (1977) (In Russian) |
| [7] | S.L. Sobolev, "Sur une généralisation de la formule de Kirchhoff" Dokl. Akad. Nauk SSSR , 1 : 6 (1933) pp. 256–262 |
| [8] | S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) |
| [9] | V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian) |
| [10] | A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) |
| [a1] | B.B. Baker, E.T. Copson, "The mathematical theory of Huygens's principle" , Clarendon Press (1950) |
| [a2] | L. Schwartz, "Théorie des distributions" , 2 , Hermann (1951) |
| [a3] | G.R. Kirchhoff, "Vorlesungen über mathematischen Physik" Ann. der Physik , 18 (1883) |
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