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 \(\lambda\) is the wave wavelength and \(y\) is the liquid depth. Show that when \(\lambda / y \ll 1\), wave speed becomes proportional to \(\sqrt{\lambda}\). In the limit as \(\lambda / y \rightarrow \infty, c=\sqrt{g y}\). Determine the value of \(\lambda / y\) for which \(c>0.99 \sqrt{g y}\).