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

120 6. Membrane and PlatesIn real applications such as loudspeakers, the amount of air displaced by themembranes, rather than the exact shape of the moving surface, determine theprinciple characteristics of generated sound waves. The radiating source can bedepicted by an equivalent simple piston of area S eq , and this piston moves througha displacement amplitude ζ eq so as to sweep the volume displacement of the actualsource. The volume displacement amplitude of the simple piston equivalent to thecircular membrane vibrating in its fundamental mode isS eq ζ eq = 0.432πa 2 A 1The nodal vibrations of actual membranes cannot be sustained with constantamplitudes because of damping forces occasioned by internal friction and externalforces associated with the radiation of acoustic energy. The amplitude of eachmode tends to decay exponentially with time as e −β nt , where β n represents thedamping constant for mode n. This damping constant generally increases withfrequency, with the result that higher frequencies damp out more quickly thandoes the fundamental.6.6 Application of Membrane Theory to the KettledrumIn addition to the damping forces mentioned above, other forces may act on amembrane and affect its vibration. The kettledrum is an example of the case ofa membrane that covers a closed space in which changes of pressures occur asthe entrapped volume of air changes in pressure incurred by the vibration of thedrumhead. A similar situation occurs with the air entrapped behind the diaphragmof a condenser microphone.The kettledrum consists of a membrane stretched taut over the open end ofa hemispherical shell. As this membrane (i.e., the drumhead) vibrates, the aircontained inside the shell undergoes alternative compressions and rarefactions.With the radial velocity of the transverse waves being considerably less than thespeed of sound in air, the pressure arising from the alternative compression anddecompression of the entrapped air is fairly uniform across the entire drumheadand depends only on the average displacement 〈z〉. With the radius of the drumheaddesignated by a, the incremental or displaced volume of the enclosed air is givenby dV = πa 2 〈z〉. Let us denote by V 0 the equilibrium or quiescent volume of theair enclosed in the kettledrum. The corresponding unperturbed pressure is P 0 . Thevibration of the air enclosed in the kettledrum is essentially an adiabatic process,with the result that the instantaneous pressure P and volume V are related to thequiescent values byPV γ = P 0 V γ0= constant (6.26)Here γ is the ratio of c p , the specific heat of the contained air at constant pressure,to c v , the specific heat at constant volume. Differentiating Equation (6.26) yields