Problem 6.14#
We start with relationship for the standing wave in 6.114,
\[\begin{equation*}
\sin(\omega \Delta t) = \beta \ell \Delta k \Delta t = \beta\, k_\ell\, \Delta t \quad.
\end{equation*}\]
Next, take the derivative with respect to the wavenumber, \(k\), yielding
\[\begin{equation*}
\frac{\partial\omega}{\partial k} \Delta t \, \cos(\omega \Delta t) = \beta \ell \Delta k \Delta t \quad,
\end{equation*}\]
which reduces to
\[\begin{equation*}
\frac{\partial\omega}{\partial k} = \frac{ \beta\, k_\ell}{\cos(\omega \Delta t)} \quad.
\end{equation*}\]