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

6.7 Forced Vibrations of a Membrane 123Assume a steady-state solutionz = Ψe iωt (6.32)which is then inserted into Equation (6.31), resulting in∇ 2 Ψ + k 2 Ψ =Pρ s c =−P (6.33)2 Twhere k = ω/c. In this situation of a driven membrane the angular frequency ωmay have any value, and the wave number k is thus not limited to discrete sets ofvalues which prevail in freely vibrating membranes.The solution to Equation (6.33) consists of two parts, one being a general solutionof the homogeneous equation 2 Ψ h + k 2 Ψ h = 0 and the second being theparticular solution Ψ p =−P/(k 2 T ). Then the complete solution can be written asΨ = AJ 0 (kr) −Pk 2 TThe immobility of the membrane at the rim r = a provides the boundary conditionΨ(a) = 0, andA = 1J 0 (ka) · Pk 2 TThe displacement of the membrane becomesz(r, t) =Pk 2 T[J0 (kr)J 0 (ka) − 1 ]e iωt (6.34)with the corresponding amplitude of the displacement at any position in the membranegiven byΨ(r) = P [ ]J0 (kr) − J 0 (ka)(6.35)T k 2 J 0 (ka)From Equation (6.35) it is seen that the amplitude of the displacement is directlyproportional to the driving force P and inversely proportional to the tensionT . The vibrational amplitude at any location on the membrane depends on thetranscendental terms enclosed by the square bracket in Equation (6.35). But ifthe driving frequency ω corresponds to any of the free-oscillation frequencies ofEquation (6.22), the overtones, the Bessel function J 0 (ka) assumes zero values,presaging infinite amplitudes. But damping forces occur in real cases, and thesemay be represented in Equation (6.31) by a damping factor –(R/ρ s )(∂z/∂t) thatlimits the amplitudes to finite maximum values.The average displacement 〈z〉 s of the driven membrane is found by averagingover the surface area of the membrane:∫ a[ ]2πe iωt P J0 (kr)0 k〈z〉 s =2 T J 0 (ka) − 1 rdrπa 2= P J 2 (ka)k 2 T J 0 (ka) eiωt (6.36)