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viewnote11 - PHYS 302 Unit 11 Viewing Notes Athabasca...

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PHYS 302 : Unit 11 Viewing Notes – Athabasca University 1 PHYS 302: Vibrations and Waves – Unit 11 Viewing Notes In all examples of explicit propagating waves thus far, the medium has been non-dispersive , which means characterized by a single speed of wave propagation. A number of important media are non-dispersive. For example, sound in air is characterized by a single speed. However, in some cases, waves are subject to dispersion , and the speed of propagation may depend on the wavelength or frequency of the wave. In fact, one case already discussed —beads on a string—does show dispersion (particularly that of five beads in Unit 7). At the time we discussed it, we had not progressed to travelling waves, so we solved it purely as a case of a normal mode or standing wave. Recall that for n beads there exist n modes, and their frequencies are 0 2( 1) 2 sin n n N . These correspond to standing waves with displacement , , cos p n p n n y A t at the position of particle p . The amplitude for that particle is , 1 sin pn p n n N A C , where C n is the amplitude associated with mode n . Although each particle has its own amplitude, C n is the same for all particles in that mode. It determines the overall amplitude of the sine curve on which the particles lie as they oscillate in unison at the frequency ω n . In Unit 8 we saw that standing waves correspond to travelling waves coming from opposite directions. In the lecture, Dr. Lewin makes the connection that allows us to re-analyze this problem to find that those travelling waves are actually dispersive. The case of N =5 with beads of mass M separated by l to give total length L is examined, and we are reminded that 0 2( 1) 2 sin n n N (2:10), with 0 T Ml . If the sin is made equal to its maximum possible value of 1, it is clear that the maximum possible frequency is ω max =2 ω 0 . (Note, however, that this does not correspond to a physically possible motion since at this frequency all the amplitudes would be 0, as described on p. 142 of French. You may also see this by putting n=N+1 in the amplitude equation above). At time 3:00, Dr. Lewin makes an important conceptual leap that likely needs elaboration. He states that the boundary conditions require n n L k . We will justify this to a greater degree than is done in the lecture. Recall that the preferred, symmetric form of the wave equation solution is ( , ) sin( ) y x t A kx t for the wave propagating toward - x . (Recall that travelling waves moving in opposite directions are needed to make a standing wave or normal mode, but here we will only consider one of them). To make the normal mode more similar to a wave, we combine the results for normal modes repeated above to get , 1 sin cos pn p n n n N y C t . We note that the time dependence is the same as in the wave solution, and we can connect the position via x if we express the bead position as a fraction of L . Since the string has length L , and this corresponds to N
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