Wave equation

The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics.

Not to be confused with Wave function.

This article focus on two-way waves in classical physics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation, which is much easier to solve and also valid for inhomogeneous media. Quantum physics uses an operator-based wave equation often as a relativistic wave equation.

$c$ is a fixed non-negative coefficient.

real

$u$ : scalar field representing a displacement from rest situation – it could be gas pressure above or below normal, or the height of water in a pond above or below rest, or another variable such as potential.

$x$ , $y$ and $z$ : represents each of the dimensions of space or position.

$t$ : represents time.

${\frac {\partial ^{2}u}{\partial t^{2}}}$ : measures how forcefully the displacement is being changed.

${\frac {\partial ^{2}u}{\partial x^{2}}}$ : measures how the displacement is varying at the point $x$ in the $x$ dimension. It is not the rate at which the displacement is changing across space but, in fact, the rate at which the change itself is changing across space – its second derivative. In other words, this term shows how the displacement's changes are squashed up near $x$ .

${\frac {\partial ^{2}u}{\partial y^{2}}}$ and ${\frac {\partial ^{2}u}{\partial z^{2}}}$ are for the other dimensions, like the term with $x$ above.

The (two-way) wave equation is a hyperbolic partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions $u = u (x, y, z, t)$ of a time variable $t$ (a variable representing time) and one or more spatial variables $x, y, z$ (variables representing a position in a space under discussion). At the same time, there are vector wave equations describing waves in vectors such as waves for an electrical field, magnetic field, and magnetic vector potential and elastic waves. By comparison with vector wave equations, the scalar wave equation can be seen as a special case of the vector wave equations; in the Cartesian coordinate system, the scalar wave equation is the equation to be satisfied by each component (for each coordinate axis, such as the x component for the x axis) of a vector wave without sources of waves in the considered domain (i.e., space and time). For example, in the Cartesian coordinate system, for $(E_{x},E_{y},E_{z})$ as the representation of an electric vector field wave ${\vec {E}}$ in the absence of wave sources, each coordinate axis component $E_{i}$ (i = x, y, z) must satisfy the scalar wave equation. Other scalar wave equation solutions $u$ are for physical quantities in scalars such as pressure in a liquid or gas, or the displacement along some specific direction of particles of a vibrating solid away from their resting (equilibrium) positions.

The scalar wave equation is

An explanation of the terms and notation:

The equation states that at any given instance, at any given point, the acceleration of the displacement is proportional to the way the displacement's changes are squashed up in the surrounding area. In other words, a more pointy displacement gets pushed back more forcefully.

Using the notations of Newtonian mechanics and vector calculus, the wave equation can be written compactly as

where the double dot on ${\ddot {u}}$ denotes two time derivatives of $u$ , $\nabla$ is the nabla operator, and $\nabla 2 = \nabla \cdot \nabla$ is the (spatial) Laplacian operator (not vector Laplacian):

Another compact notation sometimes used in physics is

A solution to this (two-way) wave equation can be quite complicated. Still, it can be analyzed as a linear combination of simple solutions that are sinusoidal plane waves with various directions of propagation and wavelengths but all with the same propagation speed $c$ . This analysis is possible because the wave equation is linear and homogeneous, so that any multiple of a solution is also a solution, and the sum of any two solutions is again a solution. This property is called the superposition principle in physics.

The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments.

$u\in C^{2}{\big (}\mathbf {R} ^{n}\times [0,\infty ){\big )}$ ,

$u_{tt}-\Delta u=0$ in $\mathbf {R} ^{n}\times (0,\infty )$ ,

$\lim _{(x,t)\to (x^{0},0)}u(x,t)=g(x^{0})$ ,

$\lim _{(x,t)\to (x^{0},0)}u_{t}(x,t)=h(x^{0})$ .

Wave equation for inhomogeneous media, three-dimensional case[edit]

For one-way wave propagation, i.e. wave are travelling in a pre-defined wave direction ( $+c$ or $-c$ ) in inhomogeneous media, wave propagation can also be calculated with a tensorial one-way wave equation (resulting from factorization of the vectorial two-way wave equation), and an analytical solution can be derived.^[9]

Other coordinate systems[edit]

In three dimensions, the wave equation, when written in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equation.

Further generalizations[edit]

Elastic waves[edit]

The elastic wave equation (also known as the Navier–Cauchy equation) in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion:

M. F. Atiyah, R. Bott, L. Garding, "", Acta Mathematica, 124 (1970), 109–189.

Lacunas for hyperbolic differential operators with constant coefficients I

M. F. Atiyah, R. Bott, and L. Garding, "", Acta Mathematica, 131 (1973), 145–206.

Lacunas for hyperbolic differential operators with constant coefficients II

R. Courant, D. Hilbert, Methods of Mathematical Physics, vol II. Interscience (Wiley) New York, 1962.

L. Evans, "Partial Differential Equations". American Mathematical Society Providence, 1998.

"", EqWorld: The World of Mathematical Equations.

Linear Wave Equations

"", EqWorld: The World of Mathematical Equations.

Nonlinear Wave Equations

William C. Lane, "", Project PHYSNET.

MISN-0-201 The Wave Equation and Its Solutions

by Stephen Wolfram and Rob Knapp, Nonlinear Wave Equation Explorer by Wolfram Demonstrations Project.

Nonlinear Wave Equations

Mathematical aspects of wave equations are discussed on the Archived 2007-04-25 at the Wayback Machine.

Dispersive PDE Wiki

Graham W Griffiths and William E. Schiesser (2009). . Scholarpedia, 4(7):4308. doi:10.4249/scholarpedia.4308

Wave equation

real

Wave equation for inhomogeneous media, three-dimensional case[edit]

Other coordinate systems[edit]

Further generalizations[edit]

Elastic waves[edit]

Lacunas for hyperbolic differential operators with constant coefficients I

Lacunas for hyperbolic differential operators with constant coefficients II

Linear Wave Equations

Nonlinear Wave Equations

MISN-0-201 The Wave Equation and Its Solutions

Nonlinear Wave Equations

Dispersive PDE Wiki

Linear and nonlinear waves