A string is stretched between \(x=0\) and \(x=L\) and has a variable density \(ho=ho_{0}+\varepsilon x\), where \(ho_{0}\) and \(\varepsilon\) are constants. The initial displacement is \(f(x)\), and the string is released from rest.

(a) If the tension \(T\) is constant, then show that the governing equation of motion is

\[ T \frac{\partial^{2} y}{\partial x^{2}}=ho \frac{\partial^{2} y}{\partial t^{2}} \]

for \(00\), with the boundary conditions \(y(0, t)=0, y(L, t)=0\), and the initial conditions \(y(x, 0)=f(x), d y(x, 0) / d t=0\).

(b) Show that the frequencies of normal mode vibration are given by \(f_{n}=\omega_{n} / 2 \pi\), where \(\omega_{n}\) are the positive roots of

\[ J_{1 / 3}(\alpha \omega) J_{-1 / 3}(\beta \omega)=J_{1 / 3}(\beta \omega) J_{-1 / 3}(\alpha \omega) \]

where

\[ \begin{aligned} & \alpha=\frac{2 ho_{0}}{3 \varepsilon} \sqrt{\frac{ho_{0}}{T}} \\ & \beta=\frac{2\left(ho_{0}+\varepsilon L\right)}{3 \varepsilon} \sqrt{\frac{ho_{0}+\varepsilon L}{T}} \end{aligned} \]