**Piano string. The figure shows data from a study of piano strings that satisfy the dispersion relation in Eq. (1.20) of the handout. (a) Use Fig. 1 to determine numerical values for 7/u and in Eq. (1.20). [Hint: At one point in your calculation for , you need to know the wavenumber. Estimate it using the approximation k& ~ kg = w/cy, where y is the velocity the wave would have if it was non-dispersive.] (b) Plot the dispersion relation w(k) for the values of 7/u and you have found. Mark on your graph a value of v, and v, that is reasonably consistent with Fig. 1. (c) On theoretical grounds, the authors of the study expected to be Ys? 4o, where Y = 1.95 x 10 N/m?, S = 9.5 x 107" m?, and o = 5.01 x 1072 kg/m. (1) Use this formula to check your numerical answer for part (a). (ii) Also confirm that the units of in parts (a) and (c) agree. (d) The fundamental transverse mode occurs at 55 Hz but the fundamental longi- tudinal mode occurs at 750 Hz. How do the restoring forces for the transverse and longitudinal mode compare? 160 7(6) 20 A,(55 Hz) 150 Vg 140 dB VELOCITY (M/S) SOUND PRESSURE LEVEL Up 130 120 0.5 1.5 - 60 FREQUENCY (XHz) Hz FREQUENCY 10090 Figure 1: (Left) Measured phase and group velocities of the Al string as a function of frequency. (Right) Fourier pressure amplitude spectrum for longitudinal excitation of the Al string. The units of amplitude are decibels; a 20 dB change means the amplitude has changed by a factor of ten.Actual continuous strings are dispersive. For example, a piano string is ap- proximately described by the dispersion relation W = KT/ M + ak2, (1.20)