91Ó°ÊÓ

Consider vibrating strings of uniform density \(\rho_{0}\) and tension \(T_{0}\). * (a) What are the natural frequencies of a vibrating string of length \(L\), fixed at both ends? *(b) What are the natural frequencies of a vibrating string of length \(H\), which is fixed at \(x=0\) and "free" at the other end (i.e., \(\partial u / \partial x(H, t)=0) ?\) Sketch a few modes of vibration as in Fig. 4.4.1. (c) Show that the modes of vibration for the odd harmonics (i.e., \(n=1,3,5\), \(\ldots\) ) of part (a) are identical to modes of part (b) if \(H=L / 2\). Verify that their natural frequencies are the same. Briefly explain using symmetry arguments.

Consider a slightly damped vibrating string that satisfies $$ \rho_{0} \frac{\partial^{2} u}{\partial t^{2}}=T_{0} \frac{\partial^{2} u}{\partial x^{2}}-\beta \frac{\partial u}{\partial t} $$ (a) Briefly explain why \(\beta>0\). *(b) Determine the solution (by separation of variables) which satisfies the boundary conditions $$ u(0, t)=0 \quad \text { and } \quad u(L, t)=0 $$ and the initial conditions $$ u(x, 0)=f(x) \quad \text { and } \quad \frac{\partial u}{\partial t}(x, 0)=g(x) \text {. } $$ Assume that this frictional coefficient \(\beta\) is relatively small \(\left(\beta^{2}<4 \pi^{2} \rho_{0} T_{0} / L^{2}\right)\).