Math 6410 • ODE

The small longitudinal free vibrations of an elastic bar are governed by the following
equation:
\[
  \rho(x)\frac{\partial^2u}{\partial t^2}
  =\frac{\partial}{\partial x}\left[E(x)\frac{\partial u}{\partial x}\right].
\]
Here \(u\) is the longitudinal displacement, \(\rho\) is the linear density of the material,
and \(E\) is its Young's modulus. Assume that the bar is constructed by welding together
two bars of different (constant) Young's moduli \(E_1,E_2\) and densities
\(\rho_1,\rho_2\), respectively. The displacement \(u\) is continuous across the junction,
which is located at \(x=0\).
\begin{enumerate}
\item[(a)] Give a weak formulation of the global initial value problem, and use this
to derive the following jump condition:
\[
  E_1u_x(0-,t)=E_2u_x(0+,t),\qquad t>0.
\]
\item[(b)] Let \(c_j^2=E_j/\rho_j\). A left incoming wave
\(u_I(x,t)=e^{i(t-x/c_1)}\) produces at the junction a reflected wave
\(u_R(x,t)=Re^{i(t+x/c_1)}\) and a transmitted wave
\(u_T=Te^{i(t-x/c_2)}\). Determine the reflection and transmission coefficients
\(R,T\), and interpret the result.
\end{enumerate}