Boussinesq approximation for water waves

Remarks

Example: for the linearised system, $\eta = A\cos(kx - \omega t)$ and $u = (\omega / (kh))\,\eta$ solve the equations when $\omega^2 = \frac{ghk^2}{1 + k^2h^2/3}$. This agrees with the exact relation $\omega^2 = gk\tanh(kh)$ up to the first correction in $kh$, since both expand as $ghk^2(1 - k^2h^2/3 + O(k^4h^4))$. Non-example: the shallow water equations, without the term $\frac{h^2}{3}u_{xxt}$, keep the nonlinearity but drop the dispersion, so they are of zeroth order in $\mu$ and not a Boussinesq system. Other choices of the velocity variable, such as the velocity at a given depth, give other Boussinesq systems. They are equivalent to the same order but differ in their higher-order behaviour. The ratio $\varepsilon / \mu = a\lambda^2/h^3$ is the Ursell number, which is of order one in the Boussinesq regime.