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

128 6. Membrane and PlatesFor the fundamental mode of vibration, the displacement of a thin circular plateis given by[ ( ) ( )]3.23.2z 1 = cos(ω 1 t + φ 1 ) A 1 J 0a r + B 1 I 0a rFrom the boundary condition relationships of Equation (6.42) the last expressionbecomes[ ( ) ( )]3.23.2z 1 = A 1 cos(ω 1 t + φ 1 ) J 0a r + 0.555I 0a rIt is interesting to observe that the amplitude at the center r = 0 is 1.0555A 1 ,not A 1 . If we compare the shape function represented by the bracketed terms onthe right-hand side of the last equation with the corresponding shape functionJ 0 (2.405r/a) for the fundamental mode of a similar-sized vibrating circular membrane,it will be found that the relative displacement of the plate near its edgeis considerably smaller than that of the membrane. Hence, the ratio of the averageamplitude to the amplitude at the center is less than that in the case for themembrane. The average displacement amplitude is given by〈Ψ 1 〉 s = 0.326A 1 = 0.309z 0where z 0 = 1.0555A 1 represents the amplitude at the center r = 0 of the plate.In the same manner we used to represent the membrane, the circular plate can bedepicted by an equivalent flat piston so thatS eq ζ eq = 0.309πa 2 z 0Plates can also undergo loaded and forced vibrations. The mathematical treatmentsof these cases are analogous to those for membranes, and the response curves aresimilar to those shown in Figure 6.4. Large amplitudes will also occur at resonancefrequencies unless there is appreciable damping.The most apparent use of the vibrating thin plate is that of the telephone diaphragms(both receiver and microphone). While these diaphragms do not providethe flatter frequency responses or frequency range of membranes in condenser microphones,they do provide adequate intelligibility, are generally far more ruggedin their construction and cheaper to manufacture. Sonar transducers used to generateunderwater sounds less than 1 kHz constitute another class of vibrating plates;the signals are produced by the variations of an electromagnetic field in an electromagnetpositioned closely to a thin circular steel plate.ReferencesFletcher, Neville H. and Rossing, Thomas D. 1998. The Physics of Musical Instruments,2nd ed. New York: Springer-Verlag, Chapter 3.Kinsler, Lawrence E., Frey, Austin R., Coppens, Alan B., and Sanders, James V. 1982.Fundamentals of Acoustics, 3rd ed. New York: John Wiley & Sons, Chapter 4.