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

124 6. Membrane and PlatesAt low frequencies ka assumes a value less than unity and the following approximationsfor Bessel functions hold true:J 0 (ka) ≈ 1 − 3(ka) 2J 2 (ka) = k2 a 2 [1 − k2 a 2 ]8 12Introducing the above approximations into Equation (6.36) yields the followingexpression for the average displacement at low frequencies,〈z〉 s ≈(1 Pa2 + k2 a 2 )(6.37)8T 6If we apply the situation as represented by Equation (6.37) to the design of a condensermicrophone, it is apparent that as long the driving frequency is sufficientlylow, i.e., ka ≪ 1, the output of the microphone will be virtually independent of thefrequency. No resonances should occur in that frequency range. The first resonanceoccurs at ka = 2.405. Becausek = 2π f ( ρS) −1/2= 2π fc Tand if we set the limiting frequency of the uniform microphone response to ka < 1,then√f < 1 T2πa ρ sThe upper frequency limit of the microphone can be elevated by either increasingthe tension T or decreasing the radius a, all other factors being equal. But thisalso has the effect of lessening the amplitude of the average displacement 〈z〉 s and,consequently, the voltage output of the microphone.When a damping factor –(R/ρ s )(∂z/∂t) is included in Equation (6.31), theresulting solution does not change except that k is replaced by k, a complexexpression represented byk 2 = ω2c 2 − iωRTThe presence of the imaginary component −ωR/T causes the average displacementto assume a finite value at resonance. Figure 6.4 displays the average displacementresponse 〈〉 s of a freely vibrating dissipationless membrane, as computedthrough the use of Equation (6.36). The amplitude assumes a value of infinity atka = 2.405. Another curve that includes the effect of damping is also plotted, andthe corresponding amplitude assumes a finite value at ka = 2.405. Both of thesecurves indicate zero responses at ka = 5.136, for which J 2 (ka) = 0. If the frequencyis increased beyond the first resonance value to approximately 1.60 timesthe first resonant frequency, a circular nodal line will appear near the rim of themembrane. As the frequency is increased, the nodular line moves inward as acircle of decreasing radius. The displacement of the membrane’s center is out of