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PHYS 302 : Unit 8 Reading Notes – Athabasca University 1 PHYS 302: Vibrations and Waves – Unit 8 Reading Notes French, A. P. Vibrations and Waves . New York: W.W. Norton & Company, 1971. Chapter 7: Progressive Waves (pp. 201–216, up to “Wave Pulses”). Chapter 6: Normal Modes of Continuous Systems. Fourier Analysis (pp. 167–170, up to “Longitudinal Vibrations of a Rod”). Chapter 7: Progressive Waves (pp. 228–230, up to “Dispersion…”). Chapter 8: Boundary Effects and Interference (pp. 253–259, up to “Impedences…”). Chapter 8: Boundary Effects and Interference (p. 264, bottom, and p. 265, up to “Waves in Two Dimensions”). Chapter 7: Progressive Waves What Is a Wave? (p. 201) According to the video lectures, we regard a wave as anything that obeys the wave equation, 22 2 1 yy vt x  . We know that any function of the form f ( x±vt ) will solve this equation. The text describes ocean waves and stresses the idea that waves carry energy from one place to another. When we think of ocean waves, we can think of a tsunami pulse in which only one peak would come across the ocean and that will obey the wave equation. But it is more usual to think of waves with many crests, which are actually periodic travelling waves. The video lectures, which take a more direct approach than the textbook, show that periodic travelling waves are well described by a function of the form (,) s i n ( ) yxt A kx t  , where we choose the sign (+ or –) depending on the direction of propagation of the wave. The textbook approaches this differently, through normal modes. The textbook approach is circular: one can relate standing waves to travelling waves by carefully applying boundary conditions. We will now look at how normal modes lead to travelling waves. As the end of this section mentions, water waves are actually complicated to analyze and are not a significant part of PHYS 302 . Normal Modes and Travelling Waves (p. 202) Fig. 7-1 shows details of a process demonstrated in the video lecture demo (with little commentary). One can imagine moving the end of a stretched rope up and down (as in (a)) to generate a travelling wave. If the rope was very long (imagine the length is infinite) the waves would continue towards the right forever. However, if there are fixed ends, there will be (negative) reflections, and if the wavelength is chosen correctly, these reflections will cause standing waves as other waves return from the reflection point. Much as the original wave is generated by the forces
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