Wave equation

A partial differential equation of the form

$$ \frac{\partial ^ {2} u }{\partial t ^ {2} } - \sum _ {k = 1 } ^ { n } \frac{\partial ^ {2} u }{\partial x _ {k} ^ {2} } = 0, $$

describing various oscillatory processes and processes of wave propagation. For the wave equation, which is an equation of hyperbolic type, two problems are usually studied: the Cauchy problem and the mixed problem.

A classical solution of the Cauchy problem, which describes wave propagation in the $ n $-dimensional Euclidean space $ E ^ {n} $, is a function $ u( x, t) $ which: is continuously differentiable in the $ ( n + 1) $-dimensional space, $ x \in E ^ {n} $, $ t \geq 0 $, is twice continuously differentiable, satisfies the wave equation in the half-space $ \{ x \in E ^ {n} , t > 0 \} $, and satisfies the initial conditions

$$ u ( x, + 0) = \phi ( x),\ \ \frac{\partial u }{\partial t } ( x, + 0) = \psi ( x), $$

where $ \phi ( x) $ and $ \psi ( x) $ are given functions.

A classical solution of the mixed problem, which describes oscillations in a bounded domain $ G \subset E ^ {n} $, is a function $ u( x, t) $ which: is continuously differentiable in the closed cylinder $ \{ x \in \overline{G}\; , t \geq 0 \} $, is twice continuously differentiable, satisfies the wave equation in the open cylinder $ \{ x \in G, t > 0 \} $, and for $ x \in G $ satisfies the initial conditions

$$ u ( x, + 0) = \phi ( x),\ \ \frac{\partial u }{\partial t } ( x, + 0) = \psi ( x) . $$

Moreover, it satisfies some boundary condition on the "lateral" surface of this cylinder.

A classical solution of Cauchy's problem for sufficiently smooth $ \phi ( x) $ and $ \psi ( x) $ is given by the so-called Poisson formula, which becomes the d'Alembert formula if $ n = 1 $. If the right-hand side of the wave equation is not zero but some given function $ f( x, t) $, the equation is called non-homogeneous and its solution is given by the so-called Kirchhoff formula. The mixed problem for the wave equation may be solved by the method of Fourier, finite-difference methods and the method of Laplace transformation.

The study of the above problems in their classical formulation given above is generalized by studies of the existence and uniqueness of classical solutions understood in a weaker sense [4], and of generalized solutions both of Cauchy's problem and the mixed problem [2], [3].

References

[1]	A.N. Tikhonov, A.A. Samarskii, "Equations of mathematical physics" , Pergamon (1963) (Translated from Russian)
[2]	S.L. Sobolev, "Partial differential equations of mathematical physics" , Pergamon (1964) (Translated from Russian)
[3]	O.A. Ladyzhenskaya, "A mixed problem for a hyperbolic equation" , Moscow (1953) (In Russian)
[4]	V.A. Il'in, "The solvability of mixed problems for hyperbolic and parabolic equations" Russian Math. Surveys , 15 : 2 (1960) pp. 85–142 Uspekhi Mat. Nauk , 15 : 2 (1960) pp. 97–154
[5]	S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian)

Comments

A more general form of the wave equation is

$$ \frac{1}{c ^ {2} } \frac{\partial ^ {2} u }{\partial t ^ {2} } - \Delta u = 0 , $$

where $ c $ (which may be a function of $ x, t $) is the speed of wave propagation.

Many classical aspects of the wave equation are discussed in [a1]. The general, modern point of view is represented in [a2].

References

[a1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[a2]	L.V. Hörmander, "The analysis of linear partial differential operators" , 3 , Springer (1985) pp. Chapts. 23–24
[a3]	J. Hadamard, "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint (1952) (Translated from French)
[a4]	F. John, "Partial differential equations" , Springer (1978)
[a5]	P.R. Garabedian, "Partial differential equations" , Wiley (1964)
[a6]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1966)
[a7]	L. Schwartz, "Mathematics for the physical sciences" , Hermann (1966)
[a8]	B.B. Baker, E.T. Copson, "The mathematical theory of Huygens' principle" , Clarendon Press (1950)
[a9]	G. Hellwig, "Partial differential equations" , Blaisdell (1964)