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The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics.
A snapshot from simulation of shallow-water equations in which shock waves are present Shallow-water equations, in its non-linear form, is an obvious candidate for modelling turbulence in the atmosphere and oceans, i.e. geophysical turbulence .
Partial chronology of FDTD techniques and applications for Maxwell's equations. [5]year event 1928: Courant, Friedrichs, and Lewy (CFL) publish seminal paper with the discovery of conditional stability of explicit time-dependent finite difference schemes, as well as the classic FD scheme for solving second-order wave equation in 1-D and 2-D. [6]
A simulation with a Boussinesq-type wave model of nearshore waves travelling towards a harbour entrance. The simulation is with the BOUSS-2D module of SMS. Faster than real-time simulation with the Boussinesq module of Celeris, showing wave breaking and refraction near the beach. The model provides an interactive environment.
Consequently, the wave equation is approximated in the SVEA as: + = . It is convenient to choose k 0 and ω 0 such that they satisfy the dispersion relation: = . This gives the following approximation to the wave equation, as a result of the slowly varying envelope approximation:
Intuitively the wave envelope is the "global profile" of the wave, which "contains" changing "local profiles inside the global profile". Each propagates at generally different speeds determined by the important function called the dispersion relation .
Once the wave equation is discretized for simulation on a computer, some small numerical reflections appear (which vanish with increasing resolution). For this reason, the PML absorption coefficient σ is typically turned on gradually from zero (e.g. quadratically ) over a short distance on the scale of the wavelength of the wave. [ 1 ]
The system of equations is closed through the description of the waves. Wave energy propagation is described through the wave-action conservation equation (without dissipation and nonlinear wave interactions): [19] [24]