Wave Equation(redirected from D'Alembert equation)
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The name given to certain partial differential equations in classical and quantum physics which relate the spatial and time dependence of physical functions. In this article the classical and quantum wave equations are discussed separately, with the classical equations first for historical reasons.
In classical physics the name wave equation is given to the linear, homogeneous partial differential equations which have the form of Eq. (1).
The simplest example of a wave equation in classical physics is that governing the transverse motion of a string under tension and constrained to move in a plane.
A second type of classical physical situation in which the wave equation (1) supplies a mathematical description of the physical reality is the propagation of pressure waves in a fluid medium. Such waves are called acoustical waves, the propagation of sound being an example. A third example of a classical physical situation in which Eq. (1) gives a description of the phenomena is afforded by electromagnetic waves. In a region of space in which the charge and current densities are zero, Maxwell's equations for the photon lead to the wave equations (3).
The nonrelativistic Schrödinger equation is an example of a quantum wave equation. Relativistic quantum-mechanical wave equations include the Schrödinger-Klein-Gordon equation and the Dirac equation. See Quantum mechanics, Relativistic quantum theory
a partial differential equation that describes the process of propagation of a disturbance in a medium. In the case of small disturbances and a homogeneous, isotropic medium, the wave equation has the form
where x, y, and z are spatial variables; t is time; u = u(x, y, z) is the function to be determined, which characterizes the disturbance at point (x, y, z) and time t; and a is the velocity of propagation of the disturbance. The wave equation is one of the fundamental equations of mathematical physics and is applied extensively. If u is a function of only two (one) spatial variables, then the wave equation is simplified and is called a two-dimensional (one-dimensional) equation. It permits a solution in the form of a“diverging spherical wave”:
u = f(t – r/a)/r
where f is an arbitrary function and . The so-called elementary solution (elementary wave) is of particular interest:
u = δ(t - r/a)/r
(where δ is the delta function); it gives the process of propagation for a disturbance produced by an instantaneous point source acting at the origin (when t = 0). Figuratively speaking, an elementary wave is an“infinite surge” on a circumference r = at that is moving away from the origin at a velocity a with gradually diminishing intensity. By superimposing elementary waves it is possible to describe the process of propagation of an arbitrary disturbance.
Small vibrations of a string are described by a one-dimensional wave equation:
In 1747, J. d’Alembert proposed a method of solving this wave equation in terms of superimposed forward and back waves: u = f(x - at) + g(x + at); and in 1748, L. Euler established that the functions f and g are determined by as-signing so-called initial conditions.
REFERENCETikhonov, A. N., and A. A. Samarskii. Uravneniia matematicheskoi fiziki, 3rd ed. Moscow, 1966.
P. I. LIZORKIN