Any general motion of two coupled oscillators can be written as a superposition of the two normal modes (symmetric and antisymmetric). A particularly instructive choice of initial conditions is the “gymnastic” one: displace only one mass and keep the other still.
In this case the resulting motion is a beat. Since the initial state contains both modes in equal measure, and the two modes have slightly different frequencies ( and ), they slowly drift out of phase and then back into phase. The observable result is that energy oscillates periodically between the two masses: first only the first one oscillates, then the energy “migrates” to the second, then returns to the first, and so on. The period of this migration is
which is longer the weaker the coupling (i.e. the closer and are).
Beats in a sentence
Two normal modes of nearby frequencies, superposed, produce a slow back-and-forth of energy. It is the same phenomenon heard as a “beat” when two nearly tuned strings sound together: here, though, the beat is spatial, the energy bounces between the two oscillators.
A classic laboratory example is that of two pendulums coupled by a string: displacing only the first, after a few oscillations the energy passes entirely to the second, then returns. The phenomenon is a prelude to the close energy-level pair of quantum mechanics.
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Topics: Oscillations and harmonic motion Concepts: Normal modes · Beats
Related exercises: Problem — Two coupled swings · Worked exercise — Two pendulums coupled by a string · Beats of two nearby frequencies