Wave dispersion and the bandwidth theorem

[RYANDTRAVERS]

Homework Statement

Consider a propagating wave packet with initial length L₀.
Use the bandwidth theorem to show that the minimum range of angular frequencies present in the wave packet is approximately:
\begin{equation}
\Delta \omega = \frac{v_{g}}{L_{0}}
\end{equation}
where v_g is the group velocity.

Homework Equations

The dispersion relationship for the wave is:
\begin{equation}
\omega ^{2} = gk
\end{equation}

The Attempt at a Solution

attached as photo along with original problem sheet. For some reason I get the answer as:
\begin{equation}
\Delta \omega = 2\pi \frac{v_{g}}{L_{0}}
\end{equation}
see method attached.

 

[Delta2]

Yes i agree with your result. but how exactly have you been taught the bandwidth theorem? I just know it as $\Delta k\Delta x\approx 2\pi$ where $\Delta k , \Delta x$ are defined properly.

 

[RYANDTRAVERS]

Well, yeah we defined it as \begin{equation} \Delta k \Delta x = 2\pi \end{equation} and then the rest can be derived from there.