$3\mathrm{.}14$ A rigid sinusoidal form is located as shown in the sketch. The form is forced to move in the $+x$ direction at speed $V.$

$($a$)$ Derive an expression for the velocity potential for the water motion induced by the moving form.

$($b$)$ Evaluate $p_c-p_t$ for the following cases:

$(1)V^2<mfrac><mrow><mi>g</mi></mrow><mrow><mi>k</mi></mrow></mfrac><mi>t</mi><mi>a</mi><mi>n</mi><mi>h</mi><mi>k</mi><mi>h</mi>$

$(2)V^2=\frac{g}{k}tanhkh$

$(3)V^2>\frac{g}{k}tanhkh$

where $p_c$ and $p_t$ denote the pressure just below the form at the crest and trough, respectively.

$($c$)$ Discuss the special significance of b$(2).$