# Kirchhoff formula

*Kirchhoff integral*

The formula

(1) |

expressing the value of the solution of the inhomogeneous wave equation

(2) |

at the point at the instant of time in terms of the retarded volume potential

with density , and in terms of the values of the function and its first-order derivatives on the boundary of the domain at the instant of time . Here is a bounded domain in the three-dimensional Euclidean space with a piecewise-smooth boundary , is the outward normal to and is the distance between and .

Let

where

The integrals and are called the retarded potentials of the single and the double layer.

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

In the case when and do not depend on , the Kirchhoff formula takes the form

and gives a solution of the Poisson equation .

The Kirchhoff formula is widely applied in the solution of a whole series of problems. For example, if is the ball of radius and centre , then formula (1) is transformed into the relation

(3) |

where

is the average value of over the surface of the sphere ,

(4) |

If and are given functions in the ball , with continuous partial derivatives of orders three and two, respectively, and is a twice continuously-differentiable function for , , then the function defined by (3) is a regular solution of the Cauchy problem (4) for equation (2) when and .

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

The Kirchhoff formula in the form

for the wave equation

(5) |

is remarkable in that the Huygens principle does follow from it: The solution (wave) of (5) at the point of the space of independent variables is completely determined by the values of , and on the sphere with centre at and radius .

Consider the following equation of normal hyperbolic type:

(6) |

with sufficiently-smooth coefficients , , , and right-hand side in some -dimensional domain , that is, a form

that at any point can be reduced by means of a non-singular linear transformation to the form

The Kirchhoff formula generalizes to equation (6) in the case when the number of independent variables is even [4]. Here the essential point is the construction of the function that generalizes the Newton potential to the case of equation (6). For the special case of equation (6),

(7) |

the generalized Kirchhoff formula is

(8) |

where is a positive number, is the piecewise-smooth boundary of an -dimensional bounded domain containing the point in its interior, and is the outward normal to . Further,

and denotes the retarded value of :

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) |

#### Comments

#### References

[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) |

**How to Cite This Entry:**

Kirchhoff formula.

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