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Continuous System

Let us now increase the number of particles N, and decrease the spacing l between them simultaneously. So that the total length L = (N + 1) l of the string remains constant. This amounts to putting all the particles closely packed. Further, we also decrease the mass of each particle so that total mass M = Nm remains constant. As N becomes infinitely large, the system tends to a continuous string of length L and mass M, fixed at both ends.

Let us put (N + 1) l = L, and m/l = mN/Nl ≃ M/L = as mass per unit length of the string; the frequency of normal modes are, therefore, given by

The displacement of a point at distance x = pl from x = 0 end is given by

The above analysis was done for transverse oscillations of N coupled particles (or a continuous string). We choose such a system as a prototype – the analysis is equally valid for longitudinal oscillations or for any other type of coupled system.