Reflect:When the pulses displace the string in the same direction they reinforce each other and give a larger pulseand when the pulses displace the string in opposite directions they tend to cancel and give a smaller pulse.12.16.Set Up:For a string fixed at each end,is the first overtone. is the second overtone and has frequency The nth harmonic is Solve: (a)(b)(c)12.17.Set Up:where The nth harmonic has frequency Solve: (a)(b)The 24thharmonic is the highest that could be heard.12.18.Set Up:where Solve: (a)(b)12.19.Set Up:For the fundamental,where xis the length of the portion of the string that is free to vibrate. The wave speed depends on the tension and linear mass density of the string, so it is the same no matterwhere the finger is placed.(a)For Then for (b)For The maximum length of the vibrating string, the distance between the bridge and the upper end of the fingerboard, is0.600 m, so this is not possible. A note cannot be played without retuning.Reflect:The speed of the waves on the string is larger than the speed of sound in air.12.20.Set Up:is the time for one cycle. For and this is the distancealong the time axis occupied by one cycle of this wave. For
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