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

Solution8.2 Lumped Acoustic Impedance 153Divide Equation (8.6) by surface area A and utilize the fact that U = dδ A, whichwill result indtZ = p (U = R + i Mω − 1 )(8.7)ωCwhereR = R rA 2 = ρ 0ck 22π , M ≡ m A 2 = ρ 0L ′A ,and the stiffness is given byC ≡ A2S = Vρ 0 c 2S = ρ 0 c 2 A2Vaccording to Equation (7.59). We also assume here a flanged resonator. In electricalterms this constitutes a RLC series circuitry, where the inductance L is the electricalanalog to M. This electrical analog to the Helmholtz resonator is illustrated inFigure 8.1. As a further aid to analytical treatment of lumped parameters, Figure 8.2summarizes the fundamental analogous elements of acoustic, mechanical, andelectrical systems. The inertance M in the acoustic system is represented by a“plug” of fluid that is sufficiently short so that all particles in the fluid can bedepicted as moving in phase under the impetus of sound pressure. The complianceC of the acoustic system is represented by an enclosed volume incorporating aFigure 8.1. The electrical analog to the Helmholtz resonator.