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

\( \hat{u} = \frac{-ikg \hat{\eta}}{-i\omega} = \frac{\cancel{-} \cancel{i} k g \hat{\eta}}{\cancel{-} \cancel{i} \omega} = \frac{k g \hat{\eta}}{\omega} \quad \text{(from } -i\omega \hat{u} = -ikg \hat{\eta} \text{)}, \quad \hat{u} = -\frac{g}{f} \frac{d\hat{\eta}}{dy} \quad \text{(from } f\hat{u} = -g \frac{d\hat{\eta}}{dy} \text{)}\)
\( \Rightarrow \quad \frac{k g \hat{\eta}}{\omega} = -\frac{g}{f} \frac{d\hat{\eta}}{dy} \quad \Rightarrow \quad \frac{d\hat{\eta}}{dy} + \frac{f k}{\omega} \hat{\eta} = 0 \quad \Rightarrow \quad \boxed{\frac{d\hat{\eta}}{dy} \pm \left( \frac{f}{c} \right) \hat{\eta} = 0} \quad \text{with } c = \sqrt{gH},\ \omega = \pm k c \) \[ \begin{aligned} &\frac{d\hat{\eta}}{dy} \pm \left( \frac{f}{c} \right) \hat{\eta} = 0 \quad \Rightarrow \quad \hat{\eta}(y) = \underbrace{\eta_0}_{\substack{\text{amplitude at the coast} \\ (y = 0)}} \cdot \underbrace{e^{-fy/c}}_{\substack{\text{decays away} \\ \text{from the coast}}} \\ &\eta(x,y,t) = \Re\left\{ \hat{\eta}(y) e^{i(kx - \omega t)} \right\} = \boxed{ \overbrace{ \eta_0 e^{-fy/c} }^{\substack{\text{amplitude and} \\ \text{decay}}} \cdot \underbrace{\cos\left(k(x - ct)\right)}_{\substack{\text{sea surface slope} \\ \text{(oscillates in }x\text{ and }t)}} } \\ &u(x,y,t) = \Re\left\{ \hat{u}(y) e^{i(kx - \omega t)} \right\} = \boxed{ \overbrace{ \eta_0 \sqrt{g/H} }^{\substack{\text{from } \partial \eta/\partial t \\ \text{and continuity}}} \cdot e^{-fy/c} \cdot \underbrace{\cos\left(k(x - ct)\right)}_{\text{velocity field}} }\end{aligned} \]