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

122 6. Membrane and PlatesTable 6.2. Allowed Frequencies of a Kettledrum.B 0 1 2 5 10k 1 a 2.405 2.545 2.68 3.02 3.485k 2 a 5.520 5.54 5.55 5.59 5.87k 3 a 8.654 8.657 8.660 8.67 8.69orwhereJ 0 (ka) =− BJ 2 (ka)(ka) 2B = γ P 0TV 0πa 2The parameter B is a dimensionless constant that compares the relative magnitudesof the restoring forces arising from the compression effects of the air trappedinside the drum and the tension applied to the drumhead. The value of B is smallif either the volume or the tension in the membrane is quite large compared withthe compressive pressure acting over the area of the membrane. In the limit whereB approaches zero, the allowed frequencies become those corresponding to thefreely vibrating circular membrane which was described earlier in this chapter.The allowed values of ka are listed in Table 6.2 for selected values of B rangingfrom 0 (which corresponds to an unimpeded vibrating circular membrane) to10 (indicative of low drum volume or light drumhead tension). The effect of theadditional term in Equation (6.28), which is proportional to the displacement andtherefore is indicative of membrane stiffness, is to elevate the allowed frequencies.The effect on the fundamental frequency is much more considerable than it is onthe higher modes of vibration. This stems from the fact that the average displacementamplitude becomes smaller with increasingly higher modes of vibration witha consequently larger number of oppositely phased segments. It is also apparentthat since pressure fluctuations inside the drum affect only the basic frequencymodes z 0n , the tonal qualities of a kettledrum can be varied by parametric changesof the drum volume V 0 and the area πr 2 of the drumhead.6.7 Forced Vibrations of a MembraneConsider a circular membrane that is acted only on one side by a evenly distributedsinusoidal driving pressure p = P cos ωt. In complex notation the pressure isgiven byp = Pe iωtand the equation of motion (6.3) becomes modified as follows:∂ 2 z∂t = 2 c2 ∇ 2 z + P e iωt (6.31)ρ s