PlanetPhysics/Wave Equation

Any wave equation describes the propagation in [[../SR/|space-time]] of a [[../CosmologicalConstant2/|wave]] (or periodic [[../CosmologicalConstant/|motion]], oscillation, `physical perturbation' or `signal') in terms of certain [[../Bijective/|types]] of [[../DifferentialEquations/|differential equations]] (such as partial differential ones); the solutions of such wave equations--usually with additonal [[../PiecewiseLinear/|boundary]] conditions-- are either propagating or stationary waves; there are numerous types of waves, and thus, there are many different types of [[../TransversalWave/|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/|heat]] transfer;

1. Laplace wave equation;

1. [[../FluorescenceCrossCorrelationSpectroscopy/|Maxwell's equations]] for electromagnetic wave propagation;

1. Schr\"odinger 'wave' equation for electrons (see also [[../QuantumHamiltonianOperator/|Hamiltonian operator]]);

1. Heisenberg's quantum [[../MathematicalFoundationsOfQuantumTheories/|dynamic]] equations (see also [[../QuantumHamiltonianOperator/|Hamiltonian operator]] and [[../QuantumHarmonicOscillatorAndLieAlgebra/|quantum harmonic oscillator and Lie algebra]]);

1. Dirac relativistic wave equation;

1. [[../Soliton/|soliton]] wave equations;

1. [[../QuarkAntiquarkPair/|spin]] wave equations;

1. [[../AlbertEinstein/|Einstein's]] gravitational wave equations;

Examples:

In its simplest form, the wave equation refers to a [[../Vectors/|scalar]] [[../Bijective/|function]] $w$ that satisfies:

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

where ${\mathrm{\nabla }}^2$ is the [[../LaplaceOperator/|Laplace operator]], and where $c$ is a fixed constant equal to the propagation [[../Velocity/|speed]] of the wave.