PlanetPhysics/Wave Equation

Any wave equation describes the propagation in space-time of a wave (or periodic motion, oscillation, `physical perturbation' or `signal') in terms of certain types of differential equations (such as partial differential ones); the solutions of such wave equations--usually with additonal boundary conditions-- are either propagating or stationary waves; there are numerous types of waves, and thus, there are many different types of wave equations. The following is a short list of such wave equations, that is however not intended to be comprehensive.

Types of Wave Equations:

1. Elastic wave equation and Hook's Law

1. Equation for sound wave propagation

1. Wave equation for heat transfer;

1. Laplace wave equation;

1. Maxwell's equations for electromagnetic wave propagation;

1. Schr\"odinger 'wave' equation for electrons (see also Hamiltonian operator);

1. Heisenberg's quantum dynamic equations (see also Hamiltonian operator and quantum harmonic oscillator and Lie algebra);

1. Dirac relativistic wave equation;

1. soliton wave equations;

1. spin wave equations;

1. Einstein's gravitational wave equations;

Examples:

In its simplest form, the wave equation refers to a scalar function $w$ that satisfies:

$\partial ^{2}(w) \over {\partial t^{2}}$ = $c^{2}\nabla ^{2}u,$

where $\nabla ^{2}$ is the Laplace operator, and where $c$ is a fixed constant equal to the propagation speed of the wave.