User:Crowsnest/Table

Properties of gravity waves on the surface of deep water, shallow water and at intermediate depth, according to linear wave theory
quantity	symbol	units	deep water ( λ < ½ h )	shallow water ( λ > 10 h )	intermediate depth ( all λ and h )
phase velocity	$\displaystyle c_p=\frac{\lambda}{T}=\frac{\omega}{k}$	m / s	$\frac{g}{2\pi} T$	$\sqrt{g h}$	$\sqrt{\frac{g\lambda}{2\pi}\tanh\left(\frac{2\pi h}{\lambda}\right)}$
group velocity	$\displaystyle c_g= c_p^2 \frac{\partial\left(\lambda/c_p\right)}{\partial\lambda}=\frac{\partial\omega}{\partial k}$	m / s	$\frac{g}{4\pi} T$	$\sqrt{g h}$	$\frac{1}{2} c_p \left( 1 + \frac{4\pi h}{\lambda}\frac{1}{\sinh\left(\displaystyle \frac{4\pi h}{\lambda}\right)} \right)$
ratio	$\displaystyle \frac{c_g}{c_p}$	-	$\displaystyle\frac{1}{2}$	$\displaystyle 1$	$\frac{1}{2} \left( 1 + \frac{4\pi h}{\lambda}\frac{1}{\sinh\left(\displaystyle \frac{4\pi h}{\lambda}\right)} \right)$
wavelength	$\displaystyle\lambda$	m	$\frac{g}{2\pi} T^2$	$T \sqrt{g h}$	for given period T, the solution of: $\displaystyle \left(\frac{2\pi}{T}\right)^2=\frac{2\pi g}{\lambda}\tanh\left(\frac{2\pi h}{\lambda}\right)$
general
angular frequency	$\displaystyle \omega$	rad / s	$\frac{2\pi}{T}$
wavenumber	$\displaystyle k$	rad / m	$\frac{2\pi}{\lambda}$