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 rélatifs à la diffusion des ondes" Acta Math. , 71 : 3–4 (1939) pp. 249–282 [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)