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

Problems for Chapter 4 878. A device that has a constant speed amplitude u(0,t) = U 0 e iωt , where U 0 is aconstant, drives a forced, fixed string.(a) Find the frequencies of the maximum amplitude of the standing wave.(b) Repeat the problem for a constant displacement amplitude y(0, t) =Y 0 e iωt .(c) Compare the results of (a) and (b) with the frequencies of mechanicalresonances for the forced fixed string. Does the mechanical amplitudecoincide with the maximum amplitude of the motion?9. Consider a string fixed at both ends, with specified values of ρ L, c, L, f, andT . Express the phase speed c ′ in terms of c and the fundamental resonancef ′ in terms of f if another string of the same materials is used but(a) the length of the string is doubled.(b) the density per unit length is doubled.(c) the cross-sectional area is doubled.(d) the tension is reduced by half.(e) the diameter of the string is doubled.10. Consider a string of length L that is plucked at the location L/3 by producingan initial displacement δ and then suddenly releasing the string. Findthe resultant amplitudes of the fundamental and the first three harmonicovertones. Draw (through computer techniques, if possible) the wave formsof these individual waves and the shape of the string occurring from the linearcombination of these waves at t = 0. Redo this problem for time t = L/c,where c represents the transverse wave velocity of the string.11. A string of length 1.0 m and weighing 0.03 kg has a mass of 0.15 kg hangingfrom it.(a) Find the speed of transverse waves in the string (Hint: neglect the weightof the string in establishing the tension in string).(b) Determine the frequencies of the fundamental and the first overtonemodes of the transverse vibrations.(c) For the first overtone of the string, compare the relative amplitude of thestring’s displacement at the antinode with that of the mass.12. A string having a linear density of 0.02 kg/m is stretched to a tension of 12 Nbetween rigid supports 0.25 m apart. A mass of 0.002 kg is loaded on thestring at its center.(a) Find the fundamental frequency of the system.(b) Find the first overtone frequency of the system.13. A standing wave on a fixed–fixed string is given by y = 3 sin (π x/4) sin 2t.The length of the string is 36 cm and its linear density 0.1 gm/cm. The unitsof x and y are in centimeters, and t is given in seconds.(a) Find the frequency, phase speed, and wave number.(b) Determine the amplitude of the particle displacement at the center of thestring and at x = L/4 and x = L/3.(c) Find the energy density for those points, and determine how much energythere is in the entire string?