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.

0.1 Types of Wave Equations:

1. Elastic wave equation and Hook’s Law
2. Equation for sound wave propagation
3. Wave equation for heat transfer;
4. Laplace wave equation;
5. Maxwell’s equations for electromagnetic wave propagation;
6. Schrödinger ’wave’ equation for electrons (see also Hamiltonian operator);
7. Heisenberg’s quantum dynamic equations (see also Hamiltonian operator and quantum harmonic oscillator and Lie algebra);
8. Dirac relativistic wave equation;
9. soliton wave equations;
10. spin wave equations;
11. Einstein’s gravitational wave equations;

0.2 Examples:

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

∂²(w)____ ∂t² = c²∇²u,

where ∇² is the Laplace operator, and where c is a fixed constant equal to the propagation speed of the wave.