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

Problems for Chapter 6 129Morse, Philip M. and Ingard, K. Uno. 1968. Theoretical Acoustics. New York: McGraw-Hill, Sections 5.2 and 5.3.Reynolds, Douglas R. 1981. Engineering Principles of Acoustics, Noise and VibrationControl. Boston: Allyn and Bacon, pp. 247–255.Wood, Alexander. 1966. Acoustics. New York: Dover Publications, pp. 429–436.Problems for Chapter 6All membranes described below may be assumed to be fixed at their perimetersunless otherwise indicated.1. Consider a square membrane, having dimensions b × b, vibrating at its fundamentalfrequency with amplitude δ at its center. Develop an expression thatgives the average displacement amplitude. Obtain a general expression forpoints having an amplitude of δ/2. Plot at least five points from this generalexpression. Do these points fall in a circle?2. A rectangular membrane has width b and length 3b. Find the ratio of the firstthree overtone frequencies relative to the fundamental frequency.3. Consider a circular membrane with a free rim. Develop the general expressionfor the normal modes and sketch the nodal patterns for the three normal modeswith the lowest natural frequencies. Express the frequencies of these normalmodes in terms of tension and surface density.4. A circular aluminum membrane of 2.5 cm radius and 0.012 cm thickness isstretched with a tension of 15,000 N/m. Find the first three frequencies of freevibration, and for each of the frequencies, determine any nodal circles.5. Prove that the total energy of a circular membrane vibrating in its fundamentalmode is equal to 0.135πρ s (aωA f ) 2 where ρ s is the area density, a the radiusof the membrane, and ω the angular frequency of the vibration, and A f thefundamental amplitude at the center.6. Steel has a tensile strength of 1.0 GPa (= 10 9 Pa) and aluminum, 0.2 GPa.Using these values as the maximum tensions, what will be the maximumfundamental frequency of a 2-cm-diameter circular membrane made up of eachof these materials? Note: for thin membranes these fundamental frequenciesare independent of the thicknesses.7. A damping force is applied uniformly over the surface of a circular membrane.This damping force per unit area =−I∂z/∂t should be introduced into theappropriate wave equation in a manner consistent with the dimensions of theterms of the equation. Solve the equation to demonstrate that the amplitudesof the free vibrations are damped exponentially as e −1/2It/ρ s.8. A kettledrum consists of a circular membrane of 50 cm diameter, with anarea density of 1.0 kg/m 2 . The membrane is stretched under a tension of10,000 N/m.(a) Determine the fundamental frequency of the membrane without a backingvessel.