handin06 - Physics 212 Classical and Modern Physics Spring...

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Physics 212 Classical and Modern Physics Spring 2008 Solutions for Hand-In Set #6 Problem A35 — Annoying your roommate (a) The whistle has an opening near the mouthpiece, so that end of the tube is an anti-node for the dis- placement of air. The end where the plunger is makes a node, so we’re looking for node/anti-node end con- ditions. Playing the whistle softly gives the longest wavelength mode: L (b) Blowing harder switches from the fundamental ( n = 1) to the first harmonic ( n = 2), which looks like this: L In the first case, there was one-quarter of a wavelength, so 1 4 λ 1 = L λ 1 = 4 L Now we have three-quarters of a wavelength, so 3 4 λ 2 = L λ 2 = 4 3 L Since the wavelength decreased by a factor of three, the frequency f = v/λ increased by a factor of three. (c) Blowing harder until it squawks unpleasantly gets us to the n = 3 mode: L Here we have a full wavelength and a quarter, so 5 4 λ 2 = L λ 2 = 4 5 L This is one-fifth the wavelength in part (a), so the fre- quency is five times the fundamental. Problem A37 — Stringed instruments (a) The three longest standing wave patterns are illus- trated in the figure. L n =1 n =2 n =3 (b) For the n = 1 mode, λ 1 = 2 L = 2 × 1 . 2 m = 2 . 4 m. The wavenumber is k 1 = 2 π λ 1 = 2 π 2 . 4 m = 2 . 62 m - 1 .
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