Problem 9-1: Transverse vibrations of a string
Derive the equation of motion for transverse vibrations in a string of mass density \(\mu\) with a tension \(T\). Show that the Lagrangian density is given by:
\begin{align*}
\lagd=\frac{1}{2}\left[\mu\left(\frac{d\eta}{dt}\right)^2-T\left(\frac{d\eta}{dx}\right)^2\right]
\end{align*}
where the string is stretched in the \(x\) directions and \(\eta(x,t)\) correspond to displacement along \(y\) of the point on the string at position \(x\).
Problem 9-2: Transverse and longitudinal vibrations of a string
Derive the equation of motion for a string of mass density \(\mu\) with a tension \(T\), and Young’s modulus, \(Y\), undergoing both transverse and longitudinal vibrations.