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PHYS 302 : Unit 7 Reading Notes – Athabasca University 1 PHYS 302: Vibrations and Waves – Unit 7 Reading Notes French, A. P. Vibrations and Waves . New York: W.W. Norton & Company, 1971. Chapter 5: Coupled Oscillators and Normal Modes (pp. 135–152). Chapter 6: Normal Modes of Continuous Systems. Fourier Analysis (pp. 161–167, up to “The Superposition of Modes on a String”). Chapter 5: Coupled Oscillators and Normal Modes Many Coupled Oscillators (p. 135) While there are interesting and important mechanical systems with one or few oscillators (e.g., a grandfather clock), waves are important in continuous systems such as liquids, solids, and gases. Thus, there is a strong motivation to increase the number of coupled oscillators we can study and to push this number towards infinity. N Coupled Oscillators (p. 136) Although the textbook discusses longitudinal oscillations, the video lecture demonstrates compound pendulums with transverse oscillation. Thinking about transverse oscillation is not hard. The procedure at the top of page 137 involves expanding the cosine in powers and then ignoring any terms of higher order than the linear term. Since the terms are less than 1, their higher powers are indeed smaller than the linear term. Ignoring higher terms (higher orders) is often called linearization . The rest position of a stretched string would be a straight line. Tension creates transverse displacement, which makes forces that try to restore the string to the rest position. When any effects due to tension changes are neglected because they are second order, these forces are transverse (in the y direction). Eq. 5-16 results from applying Newton’s second law, F=ma, to the p th particle, with the transverse acceleration written as 2 2 dt y d p and divided through by m . On page 138, in Fig. 5-11 and the discussion above it, think about the forces in the N =2 case. There must be a larger force between two particles on opposite sides of the x axis (Fig. 5-11(c)) than between two particles on the same side of the axis (Fig. 5-11(b)). It is not surprising that with a larger force, the accelerations and velocities will be larger.
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## This note was uploaded on 02/18/2011 for the course PHYS 320 taught by Professor Martinconner during the Spring '10 term at Open Uni..

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