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N Coupled Oscillator

Let us consider a one-dimensional chain of N identical particles, each of mass m, coupled together by a flexible massless string at equal distance l. The extreme ends of the string are held fixed at position x₀ = 0 and x_N+1 = ( N + 1)l. The position of p^th particle from left (x₀ = 0) is given by x_P = pl. The initial tension in the string is T.

We consider the transverse oscillations of the particles about their equilibrium positions. The amplitude of oscillations is assumed to be small so that any increase in the tension of the string as the particles oscillate is neglected.

Transverse oscillations of such a linear array of coupled oscillators serve as a prototype – the results obtained here are valid also for the longitudinal oscillations of an equivalent system.

Suppose at any instant of time t, displacement of pth particle is y_p; the displacements of adjacent particles are y_{p – 1} and y_{p + 1} as shown in fig. The p^th particle is acted upon the tensions T from two segments of the string – the resultant restoring force on it along y-axis is

where and are the angles as shown in fig. For small displacement, y << l, angles are so that we take the approximation.

The equation of motion of the p^th particle, therefore, is

where we have defined Note that there are p such equations, one for each of the N particles (p = 1 to N) with y₀ = 0 and y_N+1= 0.