wave_equation

Simulated evolution of the wave equation

Mathematics

The wave equation is a partial differential equation that describes a broad class of wave dynamics. This equation may be applied to seismic, electromagnetic, fluid, and stress waves among others.

The solution to the wave equation is of the form $u(x,y,t)$, where "u" denotes the variable being "waved". This may be the height above the surface of an ocean wave, the magnitude of the electric field in space, or anything else.

The equation is given by $\frac{\partial^2{u}}{\partial{t^2}} = c^2 \nabla^2u$

Qualitatively, this equation states that the acceleration of any point on the surface of the wave is proportional to the "sharpness" around that point by $c^2$. Sharp spikes in the wave will tend to flatten out over time.

Diffraction Example

Gaussian circular wave encounters two slits in a barrier. A diffraction pattern can be observed