THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

7Pipes, Waveguides, and Resonators7.1 IntroductionIn dealing with strings, bars, and membranes in Chapters 4–6, we considered relativelysimple geometric conditions. The situation becomes more complex whenthe sound waves are confined in a restricted amount of space. For example, whensound propagates inside a rigid-walled pipe with a wavelength that exceeds theradius of a rigid-wall pipe, the acoustic propagation inside the pipe becomes fairlyplanar. The resonance properties of the pipes driven at one end and closed off atother end constitute the basis for measuring acoustical impedances and absorptionproperties of materials. In our study of pipes we establish the models for physicalanalyses of wind musical instruments, organ pipes, and ventilation ducts (Fletcherand Rossing, 1991). In larger spaces, where the dimensions may exceed wavelengths,two- and three-dimensional standing waves can occur. We shall treat thesimple case of a waveguide with a uniform cross section, establish the conceptof group speed and phase speed which occurs with a wave propagating inside awaveguide. The acoustic waveguide is very much analogous to the electromagneticwaveguides, and it finds applications in surface-wave delay lines and in thepropagation of sound in ocean and atmospheric layers. We shall also consider thephysics of a Helmholtz resonator.7.2 The Simplest Enclosed System: InfiniteCylindrical PipeThe simplest enclosed system inside which sound propagation occurs is an infinitecylindrical pipe with its axis parallel to the direction of the propagation of the planewave in the enclosed medium. The pipe wall is assumed to be rigid, perfectlysmooth, and adiabatic (i.e., no heat transfer occurs through the wall). The pipethus has no effect on the wave propagation. A pressure wave generated by a pistonmoving in the x-direction can be expressed asp(x,t) = p 0 e i(ωt−kx) (7.1)131