Editing Wave (section)

=== Wave families ===
Sometimes one is interested in a single specific wave. More often, however, one needs to understand large set of possible waves; like all the ways that a drum skin can vibrate after being struck once with a [[drum stick]], or all the possible [[radar]] echoes one could get from an [[airplane]] that may be approaching an [[airport]].

In some of those situations, one may describe such a family of waves by a function <math>F(A,B,\ldots;x,t)</math> that depends on certain [[parameter]]s <math>A,B,\ldots</math>, besides <math>x</math> and <math>t</math>. Then one can obtain different waves – that is, different functions of <math>x</math> and <math>t</math> – by choosing different values for those parameters.

[[File:Half-open pipe wave.gif|class=skin-invert-image|thumb|right|Sound pressure standing wave in a half-open pipe playing the 7th harmonic of the fundamental (''n'' = 4)]]
For example, the sound pressure inside a [[recorder (musical instrument)|recorder]] that is playing a "pure" note is typically a [[standing wave]], that can be written as
: <math>F(A,L,n,c;x,t) = A \left(\cos 2\pi x\frac{2 n - 1}{4 L}\right) \left(\cos 2\pi c t\frac{2n - 1}{4 L}\right)</math>
The parameter <math>A</math> defines the amplitude of the wave (that is, the maximum sound pressure in the bore, which is related to the loudness of the note); <math>c</math> is the speed of sound; <math>L</math> is the length of the bore; and <math>n</math> is a positive integer (1,2,3,...) that specifies the number of [[standing wave|nodes]] in the standing wave. (The position <math>x</math> should be measured from the [[wind instrument|mouthpiece]], and the time <math>t</math> from any moment at which the pressure at the mouthpiece is maximum. The quantity <math>\lambda = 4L/(2 n - 1)</math> is the [[wavelength]] of the emitted note, and <math>f = c/\lambda</math> is its [[frequency]].) Many general properties of these waves can be inferred from this general equation, without choosing specific values for the parameters.

As another example, it may be that the vibrations of a drum skin after a single strike depend only on the distance <math>r</math> from the center of the skin to the strike point, and on the strength <math>s</math> of the strike. Then the vibration for all possible strikes can be described by a function <math>F(r,s;x,t)</math>.

Sometimes the family of waves of interest has infinitely many parameters. For example, one may want to describe what happens to the temperature in a metal bar when it is initially heated at various temperatures at different points along its length, and then allowed to cool by itself in vacuum. In that case, instead of a scalar or vector, the parameter would have to be a function <math>h</math> such that <math>h(x)</math> is the initial temperature at each point <math>x</math> of the bar. Then the temperatures at later times can be expressed by a function <math>F</math> that depends on the function <math>h</math> (that is, a [[operator (mathematics)|functional operator]]), so that the temperature at a later time is <math>F(h;x,t)</math>