A travelling wave gets reflected at a boundary such that the amplitude of the reflected wave is half that of the incident wave. Find the percentage of energy transmitted.

The incident and reflected waves are in same medium. For these two waves $\frac{{\mathrm{I}}_{\mathrm{i}}}{{\mathrm{I}}_{\mathrm{r}}}=\frac{{\mathrm{a}}_{\mathrm{i}}^2}{{\mathrm{a}}_{\mathrm{r}}^2}$
The ratio of intensities of the incident and reflected waves is $\frac{{\mathrm{I}}_{\mathrm{i}}}{{\mathrm{I}}_{\mathrm{r}}}=\frac{{\mathrm{a}}_{\mathrm{i}}^2}{{\mathrm{a}}_{\mathrm{r}}^2}=\frac{4}{1}\Rightarrow {\mathrm{I}}_{\mathrm{r}}=\frac{{\mathrm{I}}_{\mathrm{i}}}{4}$
This means 25% of the energy gets reflected back. 75% of the wave energy is transmitted.