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

130 6. Membrane and Plates(b) Determine the fundamental frequency for the membrane with a backingvessel that is a hemispherical bowl of 25 cm radius. The vessel isfilled with air at a pressure of 100 kPa, and γ (the ratio of specific heats)is 1.4.9. An undamped membrane of 4-cm radius has an area density of 1.6 kg/m 2and is stretched to a tension of 1200 N/m. It is driven by a uniform pressure7000 sin ωt Pa applied over the entire surface.(a) Determine and plot the amplitude of the displacement at the center as afunction of frequency ranging 0–2 kHz.(b) Compute and plot the shape of the membrane when driven by the appliedfrequency of 600 Hz and the applied frequency of 1000 Hz.10. A condenser microphone contains a circular aluminum diaphragm of 30-mmdiameter and 0.02-mm thickness. Aluminum has a maximum tensile strengthof 0.2 GPa.(a) What is the allowable maximum tension in N/m in the diaphragm?(b) What will be the fundamental frequency under these conditions?(c) What will be the displacement of the diaphragm at its center under theimpetus of a 500-Hz sound wave having a pressure amplitude of 1.5 Pa?(d) What will be the average displacement under the conditions of (c)?11. If the volume of air trapped behind the diaphragm of the condenser microphoneof the preceding problem is 2.5 × 10 −7 m 3 , by how much will the fundamentalfrequency be raised? Assume the normal air pressure to be equal to 100 kPaand γ = 1.4.12. Use integration over the surface of a circular thin plate vibrating in its fundamentalmode to show that the average displacement amplitude is 0.327A,where A denotes the displacement in the center of the plate.13. The diaphragm of a typical telephone receiver comes in the form of a circularsheet of steel, 4 cm in diameter and 0.18 mm thick.(a) What is the fundamental frequency, if the diaphragm is rigidly clamped atits rim?(b) How will this fundamental frequency change if the diaphragm thicknessis doubled?(c) What would happen to the fundamental frequency if the diameter of thediaphragm was increased by 50%?14. Find the fundamental frequency of a vibrating circular steel plate which isclamped at its rim and is of 25 cm diameter and 0.55 mm thickness.