M V

Finally, considering the string as linearly elastic, one can write the tension force as a linear function of the elongation, so that

where E A is the constant longitudinal stiffness of the string. This completes the system of nonlinear equations that govern the vibration of the string. In order for us to develop analytical solutions, these equations must be simplified.

Let us presuppose the existence of a static equilibrium solution of the string deflection so that u(x, t v(x, t 6(x, t e(x, t T( x, t

One then finds that such a solution exists and that, if u(0) = 0, T(x) = T0,

u(x) — eox, where S — i — i0 is the change in length of the string between its stretched and unstretched states.

If the steady-state tension Tq is sufficiently high, the perturbation deflections about the static equilibrium solution are very small. Thus, we can assume

where the ( (-quantities are taken to be infinitesimally small. Furthermore, from the second of Eqs. (2.2) one can determine 6 in terms of the other quantities, that is,

 u(x, t) — u(x) + u(x, t), v(x, t) — v(x, t),
0 0