Kirchhoff formula
Kirchhoff integral
The formula
![]() | (1) |
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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
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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
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where
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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:
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In the case when and
do not depend on
, the Kirchhoff formula takes the form
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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
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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
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that at any point can be reduced by means of a non-singular linear transformation to the form
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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) |
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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,
![]() |
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and denotes the retarded value of
:
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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) |
Kirchhoff formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kirchhoff_formula&oldid=14522