Wave Equation

Also found in: Dictionary, Thesaurus, Wikipedia.

Wave equation

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).

Here &ugr; is a parameter with the dimensions of velocity; r represents the space coordinates x, y, z; t is the time; and ∇2 is Laplace's operator defined by
Eq. (2). The function f( r ,t) is a physical observable; that is, it can be measured and consequently must be a real function.

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).

Here E is the electric field strength and B is the magnetic flux density; they are both vectors in ordinary space. The parameter c is the speed of light in vacuum. See Electromagnetic radiation, Maxwell's equations

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

Wave Equation


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 Wave Equation. 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.


Tikhonov, A. N., and A. A. Samarskii. Uravneniia matematicheskoi fiziki, 3rd ed. Moscow, 1966.


wave equation

[′wāv i‚kwā·zhən]
In classical physics, a special equation governing waves that suffer no dissipative attenuation; it states that the second partial derivative with respect to time of the function characterizing the wave is equal to the square of the wave velocity times the Laplacian of this function. Also known as classical wave equation; d'Alembert's wave equation.
Any of several equations which relate the spatial and time dependence of a function characterizing some physical entity which can propagate as a wave, including quantum-wave equations for particles.
References in periodicals archive ?
In consequence, the wave motion equation of the bar with material fields can also be equivalent to the cylindrical wave equation.
This longitudinal mode displacement satisfies a wave equation for [epsilon], different from the transverse mode displacement wave equation for [PSI].
He had an exceptional grasp of electromagnetism and of Maxwell's Equations; therefore, he was also aware that the structure of the wave equation for light just about forbade the existence of the ether.
In a nutshell, the wave equation (1) describes the propagation of an excitation generated by initial or boundary condition with a constant speed [c.
The linear wave equation is a simple example of a partial differential equation of second order.
Whereas leptons are the fundamental and excited eigenvectors of the Wave Equation as shown in the Appendix in Ref.
Sumit Gupta, senior director of the Tesla business at NVIDIA, said: "The 28X acceleration Spectraseis is seeing in elastic wave equation imaging underscores the tremendous benefits GPUs provide to the oil and gas industry.
Aassila (2001): Global Existence of solutions to a wave equation with damping and source terms.
Even though in (1) the author solves some q-differential equations using the Mellin transform, it seems more appropriate to use the q-Mellin transform, that is why in this paper the q-Mellin transform studied in (3) is used to solve some analogous Heat and Wave equations.
There is no need to try: the solution is that electrons are particles and the wave equation describes the probability distribution of their trajectories.