\[ \begin{aligned} &\underbrace{\frac{\partial \eta}{\partial t} + H \frac{\partial u}{\partial x} \quad \xrightarrow[\text{Assume } \eta = \hat{\eta}(y) e^{i(kx - \omega t)}]{\partial_t \to -i\omega,\; \partial_x \to ik} \quad -i\omega \hat{\eta} + ikH \hat{u} = 0}_{\text{Contributes to dispersion relation between } \omega \text{ and } k} \\[1.5ex] &\underbrace{\frac{\partial u}{\partial t} = -g \frac{\partial \eta}{\partial x} \quad \xrightarrow[\text{Assume } u = \hat{u}(y) e^{i(kx - \omega t)}]{\partial_t \to -i\omega,\; \partial_x \to ik} \quad -i\omega \hat{u} = -ikg \hat{\eta}}_{\text{Contributes to dispersion relation between } \omega \text{ and } k} \\[1.5ex] &\underbrace{fu = -g \frac{\partial \eta}{\partial y} \quad \xrightarrow[\text{Use } u, \eta = \hat{u}(y), \hat{\eta}(y) e^{i(kx - \omega t)}]{\partial_y \text{ acts only on } \hat{\eta}(y)} \quad f \hat{u} = -g \frac{d\hat{\eta}}{dy}}_{\text{Determines transverse dependence}} \end{aligned} \]