For shallow-water gravity waves propagating in a horizontally unbounded ocean, the crests are horizontal and oriented in a direction perpendicular to the direction of propagation. The absence of a transverse pressure gradient proportional to \(\boxed{\frac{\partial \eta}{\partial y}}\) resulted in oscillatory transverse flow and elliptic fluid-particle orbits.

Consider a gravity wave propagating parallel to a wall, whose presence allows non-zero $\frac{\partial \eta}{\partial y}$ that decays away from the wall. This situation permits a gravity wave in which $fu$ is geostrophically balanced by $-g\left(\frac{\partial \eta}{\partial y}\right)$ with $v = 0$. Consequently, fluid particle orbits are not elliptic but rectilinear.

Consider first a gravity wave propagating in the x-direction in a channel aligned with the x-direction. From the figure below, the fluid velocity under a crest is in the direction of wave propagation, and that under a trough is opposite the direction of propagation.