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

152 8. Acoustic Analogs, Ducts, and Filtersratio of pressure to volume velocity, is applied to treat acoustic radiation fromvibrating surfaces and the transmission of this radiation across lumped acousticelements and through ducts and horns. These two impedances are related to eachother byZ = z A(8.3)A is the area of a vibrating surface. If the vibrating surface is driven with a velocityu, with a force f acting on the fluid, the radiation impedance Z r is given by theratio of the force to the speed:Z r = f u(8.4)This type of impedance constitutes a part of the mechanical impedance Z m of thevibrating system. Radiation impedance relates to specific impedance at a surfaceas follows:Z r = z r(8.5)AIt is useful for dealing with coupling between acoustic waves and a driving surfaceor a driven load.8.2 Lumped Acoustic ImpedanceIn the use of lumped parameters, the advantage is taken of the assumption thatthe signal wavelength is larger than all principle dimensions, which allows forfurther simplification. Each acoustic parameter may be time varying, but it becomesvirtually independent of distance over the spatial extent of the device. When weconsider lumped or concentrated impedances rather than distributed impedances,we define that impedance in a segment of the acoustic system as the (complex) ratioof the pressure difference p (which propels that segment) to the resultant volumevelocity U. The unit of acoustic impedance is Pa s/m 3 , and it is often referred toas an acoustic ohm.Example Problem 1Consider a Helmholtz resonator described by differential Equation (7.62) whichis repeated below:m d2 δdt + R dδ2 rdr + Sδ = APeiωt (8.6)and recast the system as a lumped acoustic impedance with an electric analog.