By Leo L. Beranek, Tim Mellow
Acoustics: Sound Fields and Transducers is a completely up to date model of Leo Beranek's vintage 1954 booklet that keeps and expands at the original's designated acoustical basics whereas including sensible formulation and simulation tools.
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Additional info for Acoustics: Sound Fields and Transducers
67) In other words, there will be planes of zero particle velocity at points along the length of the tube whenever l is greater than l/2. Some examples of the particle velocity for l slightly greater than various multiples of l/2 are shown in Fig. 7. Two things in particular are apparent from inspection of these graphs. First, the quantity n determines the approximate number of half wavelengths that exist between the two ends of the tube. Secondly, for a fixed u~0 , the maximum velocity of the wave in the tube will depend on which part of the sine wave falls at x ¼ l.
Because the wave is propagated without change of shape, we need consider, in the steady state, only those solutions to the wave equation for which the time dependence at each point in space is sinusoidal and which have the same angular frequencies nu as the source. 43) pn ðx; tÞ ¼ pnþ eÀjun x=c þ pnÀ e jun x=c e jun t ; where the þ and À subscripts indicate the forward and backward traveling waves respectively. In the steady state, therefore, we may replace f1 and f2 of Eq. 44) n ¼ ÀN n ¼ ÀN Generally we omit writing < although it always must be remembered that the real part must be taken when using the final expression for the sound pressure that would actually be observed, for example, when making an animated plot of a sound field.
Regardless of the shape of the outward-going wave (or of the backward-traveling wave), it is propagated without change of shape. To show this, let us assume that, at t ¼ t1, the sound pressure at x ¼ 0 is f1(t1). At a time t þ t1 þ t2 the sound wave will have traveled a distance x equal to t2c m. At this new time the sound pressure is equal to p ¼ f1 ðt1 þ t2 À t2 cÞ ¼ f1 ðt1 Þ: In other words the sound pressure has propagated without change. The same argument can be made for the backward-traveling wave which goes in the Àx direction.
Acoustics: Sound Fields and Transducers by Leo L. Beranek, Tim Mellow