Solution

The new frequency can be calculated using the idea that:

\begin{equation} c = \lambda \nu \rightarrow \frac{c}{\lambda} = \nu \end{equation}

To calculate the frequency of the electromagnetic wave inside the moving dielectric, we need to calculate the velocity of the wave in the Earth frame and its wave length.

In the moving frame, the velocity of the wave inside the dielectric is $\frac{c}{n}$. Turning back to the Earth frame, the velocity becomes:

\begin{equation} c_1 = \frac{c/n - \beta c}{1 - \frac{\beta}{n}} = c \frac{1 - n \beta}{n - \beta} \end{equation}

The negative sign is because the dielectric's velocity points to the left in the Earth frame. Also, the wave length in the Earth frame is $\frac{\lambda}{n}$; then the frequency of the electromagnetic wave inside the moving dielectric is:

\begin{equation} \nu_1 = \frac{c_1}{\lambda/n} = \nu \frac{1 - n \beta}{n - \beta} \end{equation}

Then the difference is:

\begin{equation} \Delta \nu = \nu_1 - \nu = \nu \left(\frac{1 - n \beta}{n - \beta} - 1\right) \end{equation}