structural:scholar_edition, structural:reformatting, stylistic:style_transfer, information:extraction
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 echos 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 F (A, B, …; x, t) {\displaystyle F(A,B,\ldots;x,t)} that depends on certain parameters A, B, … {\displaystyle A,B,\ldots }, besides x {\displaystyle x} and t {\displaystyle t}. Then one can obtain different waves — that is, different functions of x {\displaystyle x} and t {\displaystyle t} — by choosing different values for those parameters.

For example, the sound pressure inside a recorder that is playing a "pure" note is typically a standing wave, that can be written as

The parameter A {\displaystyle A} defines the amplitude of the wave (that is, the maximum sound pressure in the bore, which is related to the loudness of the note); c {\displaystyle c} is the speed of sound; L {\displaystyle L} is the length of the bore; and n {\displaystyle n} is a positive integer (1,2,3,...) that specifies the number of nodes in the standing wave. (The position x {\displaystyle x} should be measured from the mouthpiece, and the time t {\displaystyle t} from any moment at which the pressure at the mouthpiece is maximum. The quantity λ = 4 L / (2 n − 1) {\displaystyle \lambda =4L/(2n-1)} is the wavelength of the emitted note, and f = c / λ {\displaystyle f=c/\lambda } 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 r {\displaystyle r} from the center of the skin to the strike point, and on the strength s {\displaystyle s} of the strike. Then the vibration for all possible strikes can be described by a function F (r, s; x, t) {\displaystyle F(r,s;x,t)}.

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 h {\displaystyle h} such that h (x) {\displaystyle h(x)} is the initial temperature at each point x {\displaystyle x} of the bar. Then the temperatures at later times can be expressed by a function F {\displaystyle F} that depends on the function h {\displaystyle h} (that is, a functional operator), so that the temperature at a later time is F (h; x, t) {\displaystyle F(h;x,t)}