### A note on the vibrating string

G. F. Carrier
1949 Quarterly of Applied Mathematics
In a previous paper [1] certain characteristics of the behavior of an elastic string undergoing periodic vibrations of moderately large amplitude were established. In this note, the same basic result is obtained in a manner which is mathematically more satisfactory. Furthermore, a refinement of the solution associated with the periodic motion of lowest frequency is obtained. It is convenient to postulate a material which obeys the stress-strain law Here, T0 is the rest tension, A the rest
more » ... n, A the rest cross-sectional area, E an elastic constant of the material, and T, u, v, d, are defined1 in Fig. 1 . This law is probably as close to reality as any we could postulate for the general run of elastic materials. In any event, a modified Eq. (1) introduces into the results only higher order effects. TCx+ax) Fig. 1 . If we apply the conditions of dynamic equilibrium to an element of the string as shown in Fig. 1 , we obtain (rigorously) [T cos 6}x = pAvtl . Now with t = {T -T0)/T0 , £ = wx/l, v2 = T2a2Et2/Pl2, a2 = TJEA, Eqs. (1) , (2) and (3) can be combined to give The boundary conditions u = 0 and v = 0 at J = 0 and £ = x can be replaced by [ (1 + a r)e 6 d£ = ir. Jo These are essentially the basic equations used in [1]. *Received Feb. 25, 1948. 'We note that tan 6 = ii*/(l + vx).