A water wave moves from a shallow to a deep region. The wavelength and speed in the shallow region are $3\mathrm{.}4$ cm and $5\mathrm{.}3$ cm$/$s$,$ respectively. If the wavelength in the deep region is $4\mathrm{.}3$ cm$,$ find

$($a$)$ the relative index of refraction from shallow to deep water

$($b$)$ the speed of the wave in the deep water