In classical continuum physics, a wave is a mechanical disturbance. Whether
the disturbance is stationary or traveling and whether it is caused by the
motion of atoms and molecules or the vibration of a lattice structure, a wave
can be understood as a specific type of solution of an appropriate mathematical
equation modeling the underlying physics. Typical models consist of partial
differential equations that exhibit certain general properties, e.g.,
hyperbolicity. This, in turn, leads to the possibility of wave solutions.
Various analytical techniques (integral transforms, complex variables,
reduction to ordinary differential equations, etc.) are available to find wave
solutions of linear partial differential equations. Furthermore, linear
hyperbolic equations with higher-order derivatives provide the mathematical
underpinning of the phenomenon of dispersion, i.e., the dependence of a wave's
phase speed on its wavenumber. For systems of nonlinear first-order hyperbolic
equations, there also exists a general theory for finding wave solutions. In
addition, nonlinear parabolic partial differential equations are sometimes said
to posses wave solutions, though they lack hyperbolicity, because it may be
possible to find solutions that translate in space with time. Unfortunately, an
all-encompassing methodology for solution of partial differential equations
with any possible combination of nonlinearities does not exist. Thus, nonlinear
wave solutions must be sought on a case-by-case basis depending on the
governing equation.Comment: 22 pages, 3 figures; to appear in the Mathematical Preliminaries and
Methods section of the Encyclopedia of Thermal Stresses, ed. R.B. Hetnarski,
Springer (2014), to appea