Math 6410 • ODE

Consider a string clamped at the end points \(a\) and \(b\), say, with \(u(a,t)=0\) and
\(u(b,t)=0\), where \(u(x,t)\) is the string's deviation from the horizontal rest position.
The kinetic energy of the string is
\[
  E_k(t)=\frac{1}{2}\int_a^b \rho u_t^2\dd s,
\]
where \(\rho(x,t)\) is the mass density and \(ds=\sqrt{1+u_x^2}\dd x\) is an infinitesimal
arc length. The potential energy consists of the sum of the energy due to the
stretching of the string, and the work done against a load:
\[
  E_p(t)=\int_a^b
  \left(d(\sqrt{1+u_x^2}-1)-lu\sqrt{1+u_x^2}\right)\dd x.
\]
Here \(d(x,t)\) is the string's elastic coefficient and \(l(x,t)\) is the load on the
string. The classical action for the continuum model is
\[
  J=\int_{t_1}^{t_2}[E_k(t)+E_p(t)]\dd t.
\]
\begin{enumerate}
\item[(a)] Substitute the integral expressions for \(E_k\) and \(E_p\) into the action
\(J\), and consider the variations \(u+\epsilon\psi\) such that \(\psi\) vanishes at the
end points \(a,b\) and at the initial and final times \(t_1,t_2\). Calculate the first
variation \(\delta J\).
\item[(b)] After integrating by parts terms of the form \(u_t\psi_t\) and \(u_x\psi_x\),
and using the boundary conditions for the variation \(\psi\), show that Hamilton's
principle \(\delta J=0\) yields the following PDE for \(u\):
\[
  (\rho u_t)_t-(1+u_x^2)^{-1/2}
  \left[d(1+u_x^2)^{-1/2}u_x\right]_x-l=0.
\]
\item[(c)] Show that if \(\rho,d\) are constants, there is no load (\(l=0\)), and the small
slope approximation \(\abs{u_x}\ll1\) holds, then the PDE of part (b) reduces to the
classical one-dimensional wave equation. Write down an expression for the wave speed.
\end{enumerate}