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Lecture 22

# Lecture 22 - Coupled Oscillations Coupled Oscillations The...

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oupled Oscillations Coupled Oscillations • The general motion can appear very complicated, but there is hidden simplicity in the form of wo normal modes, patterns of oscillation that have a Two normal modes, patterns of oscillation that have a fixed frequency. For the two springs, these are the symmetric and anti-symmetric modes. ach normal mode has its own natural frequency so Each normal mode has its own natural frequency, so there are two for the system above. • Any motion can then be expressed as a superposition of the two normal modes.

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nalysis Analysis • In normal mode 1, the masses move together in the same directions, x 1 =x 2 , where x i is the displacement of mass I from equilibrium. The middle spring never stretches! • So mass I obeys md 2 x i /dt 2 =-kx i , and ω S =(k/m) 1/2 . The asses basically oscillate independently. masses basically oscillate independently. • In the anti-symmetric mode, x 2 =-x 1 . If mass 1 moves x 1 to the right, it experiences a force F=-kx 1 +K(x 2 -x 1 )=- /2 (k+2K)x 1 . The frequency is then ω A =(k/m+2K/m) 1/2 . The anti-symmetric mode has a higher frequency.
emporal and Spatial oscillation Temporal and Spatial oscillation Another interesting thing happens with coupled oscillators. If we start mass 1 oscillating and mass 2 fixed, we observe that after a while mass 1 will stop and mass 2 will start. • The energy propagates from mass 1 to mass 2. • When we have an oscillation developing in time and ropagating in space we have the makings of a wave propagating in space we have the makings of a wave. • Similar phenomena would occur for 3 coupled oscillators (3 natural frequencies, 3 normal modes) or even N coupled oscillators (N natural frequencies, N normal modes). • We will take N Æ and look at waves in elastic media.

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Waves in confined media • When many oscillators hook together, oscillations can develop in space and time. We’ll confine ourselves to single frequency sine or cosine waves.
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Lecture 22 - Coupled Oscillations Coupled Oscillations The...

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