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.