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Unformatted text preview: Harmonic Waves A wave generated by the simple harmonic vibrations of the particles of a medium is called a harmonic wave . What is the equation for the displacement of a simple harmonic motion? At t = 0 if the particle is at the equilibrium position, the displacement y is: y = A sin ( ϖ t) ince a harmonic wave is formed by the varying Since a harmonic wave is formed by the varying displacements of the vibrating particles, the equation for a harmonic wave is: y = A sin ( ϖ t) 2 2 T f π ϖ π = = 2 sin y A t T π = 1 A is the amplitude of the wave and f is the frequency. The displacement y can also be written as a function of x 2 sin y A t T π = v f λ = [ ] sin 2 A ft π = λ sin 2 v y A t π λ = n 2 si vt A π λ = [ ] sin y A x k = is called 2 wave number the k π λ = 2 To describe a wave advancing to the right, we replace x by x – vt where v is the wave speed. [ ] sin y x A k = The equation for a wave advancing to the right is: y = A sin[k(x – vt)] 2 ) ( sin x vt y A π λ = 2 2 sin A t v x π π λ λ  = = A sin ( k x – 2 π f t) y = A sin (kx – ϖ t) Similarly a wave advancing to the left can be written as: y = A sin[k(x + vt)] y = A sin (kx + ϖ t) http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Fla sh/ClassMechanics/TravelWaves/TravelWaves.html 3 The wave y(x,t) = A cos k(x + 34 t) represent a traveling wave where x is in meters and t in second. What are the direction and the speed of the wave? Sine function or cosine function only determines the initial state of the wave and not the direction. Example 1 The wave y(x,t) = A sin k(x + v t) is a wave traveling to the left. The wave y(x,t) = A cos k(x + 34 t) is a wave traveling to the left with a speed equal to 34 ms1 . 4 At time t = 0, the shape of a wave pulse on a string is given by the function: where x is in meters. Plot y(x,0) versus x and give the general wave function y(x,t) at any time t if (i) the pulse is moving in the positive xdirection with a speed of 10...
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This note was uploaded on 05/01/2011 for the course PHY 2048 taught by Professor George during the Fall '10 term at Edison State College.
 Fall '10
 George
 Physics, Simple Harmonic Motion

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