Coupled Oscillator Model

Let y_i= y(x_i,t) represent the displacement of a particle with horizontal position x_i along the oscillator chain. Because it is assumed that the particles do not move in the x-direction, we need only consider forces in the vertical direction. The force F_i on the i-th particle depends on the relative vertical displacement between that particle and its nearest neighbors and can be written as

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where k is the Hook's law constant. Because the first and last particles in the lattice are fixed, the we compute particle accelerations starting with the second particle and continuing through the lattice until we reach the next-to-last particle.

One way of understanding a lattice of N coupled oscillators of length L and mass M is to study the motion of its normal modes. A normal mode is a special configuration (state) where every particle moves sinusoidally with the same frequency. The m-th mode Φ_m of the oscillator chain of lenght L is

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The system stays in a single mode and every particle oscillates with constant angular frequency ω_m if the oscillator chain is initialized in a single mode.

Normal modes are important because an arbitrary iniital configuratiopn can be expressed as sum of normal modes.