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viewnote8 - PHYS 302 Unit 8 Viewing Notes Athabasca...

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PHYS 302 : Unit 8 Viewing Notes – Athabasca University 1 PHYS 302: Vibrations and Waves – Unit 8 Viewing Notes The textbook treated normal modes immediately upon derivation of the wave equation, whereas the video lectures focused on progressive (i.e., travelling) waves. Now the two converge as we consider normal modes in continuous media. One approach could be to take the results from N beads on a string and let N go to infinity. This is easy to envision qualitatively (1:20), but quantitatively, we will use an approach based on travelling waves and boundary conditions. A few definitions are needed. The wavelength λ (pronounced lambda) is the distance a disturbance travels in one oscillation time, as it propagates with speed T v as described by 2 sin[ ( )] y A x vt . Note that this is a solution of the wave equation, since any function f ( x±Ct ) is a solution. This particular function describes a periodic wave as opposed to just a pulse. If, for a fixed time t , one advances by a distance x in the direction of propagation of a wave, the disturbance will change as a sinusoid. The periodic nature of the wave is clear, since if one advances by a distance λ , the phase will change by 2 π , so the same disturbance will repeat. If one sits at a fixed point and lets the wave move past as time t naturally increases, the pattern repeats after one period P (Note: the notation T was previously used for some other periodic motions, but here we want to retain T as the tension). Thus, if t increases by an amount P , the phase also increases by 2 π . Since the overall argument for phase is 2 ( ) x vt , for the change in phase 2 2 2 2 ( ( )) ( ) x v t P x vt vP . This immediately gives vP . We can describe this by saying that the wavelength is the distance traversed by the wave in one period (4:20). Since 2 π radians of phase go by in one period lasting time P , the rate at which phase changes (the angular frequency) is 2 P . From this, we extend vP to 2 v and this gives 2 v (4:20) (and v f ). We can introduce the wave number 2 k ; then kv . Note the discrepant definition of k mentioned relative to the French textbook Also, for your information, “Bekefi and Barrett” is a textbook used in the MIT 8.03 course but not used in this course. Use of k brings a more symmetric form of the wave equation: ( , ) sin( ) y x t A kx t . We now consider two identical waves propagating in opposite directions. The two such waves are 1 ( , ) sin( ) y x t A kx t and 2 ( , ) sin( ) y x t A kx t , which superpose by simple addition. You may recall that the sum of two sins is the sin of half the sum of the arguments times the cos of half the difference. This is analogous to the identity proved for cos on page 26 of French and mentioned in the Reading Notes for Unit 2.
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