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:

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  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:

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

  =  

where   is the Laplace operator, and where   is a fixed constant equal to the propagation speed of the wave.