Wave equation

Most famously, it can be derived for the case of a string vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension.

[2] Another physical setting for derivation of the wave equation in one space dimension uses Hooke's law.

The wave equation in the one-dimensional case can be derived from Hooke's law in the following way: imagine an array of little weights of mass m interconnected with massless springs of length h. The springs have a spring constant of k: Here the dependent variable u(x) measures the distance from the equilibrium of the mass situated at x, so that u(x) essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material.

A uniform bar, i.e. of constant cross-section, made from a linear elastic material has a stiffness K given by

Then, to solve the first (inhomogenous) equation relating v to u, we can note that its homogenous solution must be a function of the form F(x - ct), by logic similar to the above.

Expanding out the left side, rearranging terms, then using the change of variables s = x + ct simplifies the equation to

Eigenmodes are useful in constructing a full solution to the wave equation, because each of them evolves in time trivially with the phase factor

[6] The chirp wave solutions seem particularly implied by very large but previously inexplicable radar residuals in the flyby anomaly and differ from the sinusoidal solutions in being receivable at any distance only at proportionally shifted frequencies and time dilations, corresponding to past chirp states of the source.

The above vectorial partial differential equation of the 2nd order delivers two mutually independent solutions.

In spherical coordinates this leads to a separation of the radial and angular variables, writing the solution as:[10]

The angular part of the solution take the form of spherical harmonics and the radial function satisfies:

[citation needed] For physical examples of solutions to the 3D wave equation that possess angular dependence, see dipole radiation.

Although the word "monochromatic" is not exactly accurate, since it refers to light or electromagnetic radiation with well-defined frequency, the spirit is to discover the eigenmode of the wave equation in three dimensions.

Following the derivation in the previous section on plane-wave eigenmodes, if we again restrict our solutions to spherical waves that oscillate in time with well-defined constant angular frequency ω, then the transformed function ru(r, t) has simply plane-wave solutions:

From this we can observe that the peak intensity of the spherical-wave oscillation, characterized as the squared wave amplitude

Let a family of spherical waves have center at (ξ, η, ζ), and let r be the radial distance from that point.

[citation needed] The main use of Green's functions is to solve initial value problems by Duhamel's principle, both for the homogeneous and the inhomogeneous case.

[18]: 698 Hadamard's conjecture states that this generalized Huygens' principle still holds in all odd dimensions even when the coefficients in the wave equation are no longer constant.

The case where u is required to vanish at an endpoint (i.e. "fixed end") is the limit of this condition when the respective a or b approaches infinity.

The method of separation of variables consists in looking for solutions of this problem in the special form

A solution that satisfies square-integrable initial conditions for u and ut can be obtained from expansion of these functions in the appropriate trigonometric series.

The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions.

where f and g are defined in D. This problem may be solved by expanding f and g in the eigenfunctions of the Laplacian in D, which satisfy the boundary conditions.

In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary B.

Physical examples of source functions include the force driving a wave on a string, or the charge or current density in the Lorenz gauge of electromagnetism.

One method to solve the initial-value problem (with the initial values as posed above) is to take advantage of a special property of the wave equation in an odd number of space dimensions, namely that its solutions respect causality.

For the other two sides of the region, it is worth noting that x ± ct is a constant, namely xi ± cti, where the sign is chosen appropriately.

Looking at this solution, which is valid for all choices (xi, ti) compatible with the wave equation, it is clear that the first two terms are simply d'Alembert's formula, as stated above as the solution of the homogeneous wave equation in one dimension.

As an aid to understanding, the reader will observe that if f and ∇ ⋅ u are set to zero, this becomes (effectively) Maxwell's equation for the propagation of the electric field E, which has only transverse waves.

For light waves, the dispersion relation is ω = ±c |k|, but in general, the constant speed c gets replaced by a variable phase velocity: