Starting from the free$-$surface boundary conditions considering surface tension,

$($a$)$ show that in deep water, the dispersion relation is

${}^2=gk+\frac{k^3}{}$

Where $$ is the surface tension and $$ is the density of water.

$($b$)$ It can be found using the dispersion relation above, that there is a minimum phase velocity

for water waves, find the minimum phase velocity and the corresponding wavelength for

water wave, using $g=9\mathrm{.}8\frac{m}{s^2},$ and $=0\mathrm{.}074\frac{N}{m},=1000\frac{kg}{m^3}.($hint: take $\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{k}}=0)$