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Normal Modes

A normal mode of vibrations of a coupled system means that each particle of the system should oscillate with same frequency ω and in phase; the amplitudes may however be different. That is, the motion of each particle in a particular mode is given by

y_p =A_p cos (ω t + Ø) = A_p cos ω t p = 1, …., N

where we put all Ø = 0, assuming that v_p = dy_p/dt = 0 at t = 0.

Substituting above, we get

Since above equation holds for any time t, it implies

Hence, if a normal mode exists such that oscillations of each particle is governed, then the amplitudes of any three adjacent particles must be related by above equation. A possible solution of above equation shall determine what values of ω can exist.

We know that in case of stationary vibrations of a continuous string fixed at two ends, the amplitudes of a point x is given by A ( x ) = A sin kx, k = 2π/λ. We guess our solution from there: for pth particle, x_p = pl; hence, we try the solution

A_p = A sin k pl

Therefore,

A_p+1 + A_p–1 = A [sin (p + 1) kl + sin (p – 1) kl] = 2 A sin kpl cos kl

Thus, eq. would be satisfied if

The relation between k and ω is called the dispersion relation.

We have still not defined k. Its value is determined from the boundary conditions that the string is fixed at x = 0 and x = (N + 1) l. That is y₀ = 0 and y_N+1 = 0, or A_{p = 0} = 0 and A_{p = N + 1} = 0. The first condition is satisfied, the second condition implies that

sin k (N + 1) l = 0 or k (N + 1) l = n π, n = 1, 2, 3, …..