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

8Acoustic Analogs, Ducts, and Filters8.1 IntroductionIn this chapter we deal with lumped and distributed acoustic elements, applyingelectrical and mechanical analogs to acoustic behavior in order to treat differentduct geometries, acoustic filters, and networks. The reflection and transmissionof sound waves at piping interfaces, where the acoustic impedance changes, areanalogous to the behavior of current waves in a transmission line at locations wherethe electrical impedance undergoes a change.A simple mechanical system can often be converted into analogous electricalsystems and solved in analogous terms. The motion of a fluid is compared to thebehavior of current in an electrical circuit, with a pressure gradient between twopoints playing the role of voltage across the corresponding parts of the circuit.In terms of electricity the impedance is voltage divided by the current, whichcorresponds to the effect of lumped elements of inductance, capacitance, andresistance. In acoustics the acoustic impedance Z of a fluid acting with acousticpressure p on a surface of area A is given byZ = p U(8.1)where U represents the volume velocity of the fluid in the acoustic element ofinterest. U is not really a vector; it is a speed representative of a scalar quantity,unlike velocity which is a magnitude with an indicated direction. The acousticimpedance Z defined by Equation (8.1) is a complex quantity.The specific acoustic impedance z is given byz = p u(8.2)where u is the particle velocity, not the volume velocity. The specific acousticimpedance, which is useful for treating the transmission of acoustic waves fromone medium to another, is a characteristic of the propagation medium and the typeof propagating wave. The acoustic impedance, defined by Equation (8.1) as the151