The wave equation is a partial differential equation that describes the propagation of various types of waves.
The equation appears throughout many fields in physics, including acoustics, fluid dynamics, electromagnetism, and quantum mechanics. With some modifications, it can even describe the spread of traffic jams on busy highways!
The one-dimensional equation was first discovered by d’Alembert in 1746 as he studied how vibrations propagated through a string, and the two- and three-dimensional equations were solved soon after by Euler during his study of acoustics.
The simulations above show the propagation of a disturbance on a two-dimensional surface for two different sets of boundary conditions  .
Mathematica code posted here.