An examination of \(\boxed{\frac{\partial v}{\partial t} + fu = -g \frac{\partial \eta}{\partial y}}\) reveals fundamental differences between Poincaré waves away from boundaries and Kelvin waves. For a Poincaré wave, the crests are horizontal, and the absence of a transverse pressure gradient requires a \( \frac{\partial v}{\partial t} \) to balance the Coriolis acceleration, resulting in elliptic orbits. In a Kelvin wave, a transverse velocity is prevented by a geostrophic balance of \( fu \) and \( -g \frac{\partial \eta}{\partial y} \)

• The Poincaré wave solutions are produced in the presence of a height perturbation in a rotating shallow water system
• The Kelvin waves require the presence of a boundary (or the equator)
• The Rossby waves require the presence of a gradient in potential vorticity