Chapter 11, Problem 3P

Solution of the complete differential equations for wave motion without surface tension shows that wave speed is given by

$c=\sqrt{\frac{g\lambda }{2\pi }\tanh \left(\frac{2\pi y}{\lambda }\right)}$

where λ is the wave wavelength and y is the liquid depth. Show that when λ = y ≪ 1, wave speed becomes proportional to $\sqrt{\lambda }$. In the limit as λ/y → ∞, $c=\sqrt{gy}$. Determine the value of λ/y for which $c>0.99\sqrt{gy}$.