From the shallow-water set: \[ \frac{\partial \eta}{\partial t} + H \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) = 0, \quad \frac{\partial u}{\partial t} - fv = -g \frac{\partial \eta}{\partial x}, \quad \text{and} \quad \frac{\partial v}{\partial t} + fu = -g \frac{\partial \eta}{\partial y} \]

The equations of motion for a Kelvin wave propagating along a coast aligned with the x-axis are: \[ \frac{\partial \eta}{\partial t} + H \left( \frac{\partial u}{\partial x} + \cancel{\frac{\partial v}{\partial y}} \right) = 0, \quad \frac{\partial u}{\partial t} - \cancel{fv} = -g \frac{\partial \eta}{\partial x}, \quad \text{and} \quad \cancel{\frac{\partial v}{\partial t}} + fu = -g \frac{\partial \eta}{\partial y} \] \[ \frac{\partial \eta}{\partial t} + H \frac{\partial u}{\partial x} = 0, \quad \frac{\partial u}{\partial t} = -g \frac{\partial \eta}{\partial x}, \quad \text{and} \quad fu = -g \frac{\partial \eta}{\partial y} \]