Eigentone - Standing Waves

Standing Waves

A standing wave is a continuous form of normal mode. In a standing wave, all the space elements (i.e. (x, y, z) coordinates) are oscillating in the same frequency and in phase (reaching the equilibrium point together), but each has a different amplitude.

The general form of a standing wave is:

$\Psi(t) = f(x,y,z) (A\cos(\omega t) + B\sin(\omega t))$

where ƒ(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 (superposition) of waves and their reflections (although one may also say the opposite; that a moving wave is a superposition of standing waves). The geometric shape of the medium determines what would be the interference pattern, thus determines the ƒ(x, y, z) form of the standing wave. This space-dependence is called a normal mode.

Usually, for problems with continuous dependence on (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 (a discrete infinity of ) normal modes (usually numbered n = 1, 2, 3, ...). If the problem is not bounded, there is a continuous spectrum of normal modes.

Famous quotes containing the words standing and/or waves:

“Still bent to make some port he knows not where,
Still standing for some false impossible shore.”
—Matthew Arnold (1822–1888)

“Come unto these yellow sands,
And then take hands.
Curtsied when you have and kissed
The wild waves whist,
Foot it featly here and there;
And, sweet sprites, the burden bear.”
—William Shakespeare (1564–1616)