Editing Normal mode (section)

=== Standing waves ===
A [[standing wave]] is a continuous form of normal mode. In a standing wave, all the space elements (i.e. {{math|(''x'', ''y'', ''z'')}} coordinates) are oscillating in the same [[frequency]] and in [[phase (waves)|phase]] (reaching the [[mechanical equilibrium|equilibrium]] point together), but each has a different amplitude.

[[File:Standing-wave05.png]]

The general form of a standing wave is:

<math display="block">
\Psi(t) = f(x,y,z) (A\cos(\omega t) + B\sin(\omega t))
</math>

where {{math|''f''(''x'', ''y'', ''z'')}} represents the dependence of amplitude on location and the cosine/sine are the oscillations in time.

Physically, standing waves are formed by the [[Interference (wave propagation)|interference]] (superposition) of waves and their reflections (although one may also say the opposite; that a moving wave is a [[superposition principle|superposition]] of standing waves). The geometric shape of the medium determines what would be the interference pattern, thus determines the {{math|''f''(''x'', ''y'', ''z'')}} form of the standing wave. This space-dependence is called a '''normal mode'''.

Usually, for problems with continuous dependence on {{math|(''x'', ''y'', ''z'')}} there is no single or finite number of normal modes, but there are infinitely many normal modes. If the problem is bounded (i.e. it is defined on a finite section of space) there are [[countably many]] normal modes (usually numbered {{math|1=''n'' = 1, 2, 3, ...}}). If the problem is not bounded, there is a continuous spectrum of normal modes.