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AU OpenCourseWare Connors, Martin, Reading Notes on Vibrations and Waves. Athabasca, AB: Athabasca University, 2010. Please use the following citation format: Connors, Martin, Reading Notes on Vibrations and Waves. (Athabasca University: AU OpenCourseWare). (accessed [Date]). License: Creative Commons Attribution-NonCommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit:
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PHYS 302 : Unit 1 Reading Notes – Athabasca University 1 PHYS 302: Vibrations and Waves – Unit 1 Reading Notes French, A. P. Vibrations and Waves . New York: W.W. Norton & Company, 1971. Chapter 1: Periodic Motions (pp. 3-16). Chapter 3: The Free Vibrations of Physical Systems (pp. 41-62, up to “The Decay of Free Vibrations”). Chapter 1: Periodic Motions (p. 3) The introduction gives several examples of oscillatory systems, stressing the idea of periodicity : the repetition of a pattern regularly in time. The period of vibration is given the symbol T . Sinusoidal Vibrations (p. 4) Page 5 gives a general expression for a restoring force on an object near equilibrium: 2 2 dx dt mk x  . The linear term in this expression is often the most important one. Consider Hooke’s law for extension of a spring. In this law, the restoring force is F=–kx, which is linear. When we apply Newton’s second law (in the form ma=F) to such a situation, we get 2 2 dt x . The general solution, ) cos( t A x , differs from Eq. 1-1 only in notation and could be converted to the form in that equation by the substitution δ φ 0 - π /2 (recall, or show with a small sketch, that sin( θ + π /2)=cos θ . δ or φ 0 are initial phase angles, determined by the conditions at time t =0). For this to be the solution, we require k m , which is the angular frequency of the motion, since ω t is an angle (in radians). Sinusoidal vibrations arise naturally for small amplitudes. All periodic phenomena can be built up from suitable combinations of sinusoidal variations of certain frequencies, amplitudes, and phases. The Description of Simple Harmonic Motion (p. 5) Eq. 1-1 can be written 0 sin( ( )) xA t  , so that it is at time t =- φ 0 / ω that the displacement is first 0 (see Fig. 1-2). If the motion actually began (physically) at time t =0, as suggested by the dashed lines before this time, then φ 0 is the angular argument of the sin function at t =0; since the argument is called the phase , this is the initial phase . Since the amplitude of the sin function itself is 1, it must be multiplied by A to describe motion of amplitude A . When the phase increases by 2 π , the sin function repeats. To make the phase increase by 2 π , t must increase to t+2 π / ω . This corresponds to one period of the sin function, so the period T =2 π / ω . It is stated without proof that motion is specified if the initial position and velocity are given, and the general form for velocity is the time derivative of Eq. 1-1, 00 (s i n ( ) c o s ( ) d dt At A t 
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