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Vibrating String Energy

A vibrating string, stretched under tension T, possesses both kinetic energy of motion and elastic potential energy. The displacement of the string, in nth mode of vibration, is given by

The above solution holds for string fixed at both ends, i.e.

y = 0 at x = 0 and x = L

Further, if we define t = 0 as the moment when string is in its equilibrium position, i.e. y (x, t) = 0 for all x, then we get

cos Ø = 0 or Ø = π/2

Hence, we consider the solution

Now, the kinetic energy of an element dx of the string at position x is

The total kinetic energy at time t, in nth mode, is therefore

where, M = L is the mass of the string.

The instantaneous potential energy of element dx is equal to work done by tension T in stretching the given element dx of the string to length ds when the string is displaced from its equilibrium position:

dU_n = T (ds – dx)