A tuning fork of frequency f0 begins vibrating at time t = 0 and is stopped after a time interval ?t. The waveform of the sound at some later time is shown as a function of x. Let N be the (approximate) number of cycles in this waveform.

(a) How are N, f0, and ?t related?

(b) If x is the length in space of this wave packet, what is the wavelength in terms of ?x and N?

(c) What is the wave number k in terms of N and ?x?

(d) The number N is uncertain by about ?1 cycle. Use Figure to explain why.

(e) Show that the uncertainty in the wave number due to the uncertainty in N is 2p/?x.

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