The wave drag force $\backslash ($ F $\backslash )$ on a boat is a function of the boat velocity $\backslash ($ V $\backslash ),$ the boat dimensions $($i$.$e$.,$ width $\backslash ($ W $\backslash )$ and length $\backslash ($ L $\backslash )),$ fluid density $\backslash (\backslash$rho $\backslash ),$ and gravitational acceleration $\backslash ($ g $\backslash ).$ A $\backslash (1$: $5\backslash )$ scale model is used in a tow tank to study the prototype that travels at $\backslash (5\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$

$($a$)$ Using the step$-$by$-$step method, determine the $\backslash (\backslash$pi $\backslash )-$groups relating to these variables. Express one of the $\backslash (\backslash$pi $\backslash )-$terms as $\backslash (\backslash$frac$\{$V$\}\{\backslash$sqrt$\{$g L$\}\}\backslash ).$

$($b$)$ Determine the velocity of the model to ensure dynamic similarity between model and prototype.

$($c$)$ Determine the ratio of wave drag forces between model and prototype if the density of the fluid used in model testing is three times higher than the density of fluid for the prototype boat.