$1$ Refraction and Shoaling

In class, it was shown that over straight parallel depth contours $($i$.$e$.,h$ is dependant only on $x$ and $\frac{sin}{c}=$ constant, where $E$ is the energy density, $c_g$ is the group velocity, and $$ is the incidence angle.

$($a$)$ Show that the effect of variable $$ is to reduce $E{c}_g$ as the waves propagate shore$-$ward $($ decreases due to refraction$).$

$($b$)$ Show that in shallow water $($small $kh)$ at sufficiently large incidence angles $($e$.$g$.,$ close to $90$ degree$)$ the refraction effect overcomes the shoaling effect.