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

86 4. Vibrating StringsKinsler, Lawrence E., Frey, Austin R., Coppens, Alan B., and Sanders, James V. 1999.Fundamentals of Acoustics, 4th ed. New York: John Wiley & Sons, Chapter 2. (Still agood textbook which dates back to 1950.)Morse, Philip M. 1982 (reprint from 1948 edition published by McGraw-Hill). Vibrationand Sound. Woodbury, NY: Acoustical Society of America, Chapter 3.Morse, Philip M. and Ingard, K. Uno. 1968. Theoretical Acoustics. New York: McGraw-Hill, Chapter 4.Reynolds, Douglas R. 1981. Engineering Principles of Acoustics, Noise and VibrationControl. Boston: Allyn and Bacon, pp. 213–224.Problems for Chapter 41. Show by direct substitution that each of the following expressions constitutessolutions of the wave equation:(a) f (x − ct)(b) ln [ f (x − ct)](c) A(ct − x) 3(d) sin [A(ct − x)]2. Show which of the followings are solutions and not solutions to the waveequation:(a) B(ct − x 2 )(b) C(ct − c)t(c) A + B sin(ct + x)(d) A cos 2 (ct − x) + B sin(ct + x)3. Plot (by computer if possible) the expression y = Ae −B(ct−x) for times t =0 and t = 1.0, with A = 6 cm, B = 4cm −1 , and c = 3cm/s. Discuss thephysical significance of these curves.4. Consider a string of density 0.05 g/cm, in which a wave form y = 4 cos (5t −3x) is propagating. x and y are expressed in centimeters, and time t in seconds.(a) Determine the amplitude, phase speed, frequency, wavelength, and thewave number.(b) Find the particle speed of the string element at x = 0 at time t = 0.5. A string is stretched with tension T between two rigid supports located atx = 0 and x = L. It is driven at its midpoint by a force F cos ωt.(a) Determine the mechanical impedance at the midpoint.(b) Establish that the amplitude of the midpoint is given by F tan (kL/2)/(2kT).(c) Find the amplitude of displacement at the quarter point x = L/4.6. Determine the mechanical impedance with respect to the applied force drivinga semi-infinite string at a distance L from the rigid end. What is the significanceof the individual terms in the expression for mechanical impedance?7. Consider a string of density 0.02 kg/m that is stretched with a tension of 8 Nfrom a rigid support to a device producing transverse periodic vibrations at theother end. The length of the string is 0.52 m. It is noted that for a specific drivingfrequency, the nodes are spaced 0.1 m apart and the maximum amplitude is0.022 m. What are the frequency and the amplitude of the driving force?