Equations of Motion

In the northern hemisphere, the surface must slope as indicated in the figure, that is, downward to the left under the crest and upward to the left under the trough, so that the pressure force has the current directed to its right. The result is that the amplitude of the wave is larger on the right-hand side of the channel, looking in the direction of propagation. The current amplitude, like the surface displacement, also decays to the left.

If the left wall is moved away to infinity, what remains is a gravity wave trapped to the coast. Such coastally-trapped long gravity waves, in which the transverse velocity \( v = 0 \) everywhere, are called Kelvin waves

Such waves can propagate only in a direction such that the coast is to the right (looking in the direction of propagation) in the northern hemisphere and to the left in the southern hemisphere. The opposite direction of propagation would result in a sea surface displacement increasing exponentially away from the coast, which is not possible.

In each vertical plane parallel to the coast, the currents (shown by arrows) are entirely within the plane and are exactly the same as those for a long gravity wave in a nonrotating channel. However, the surface elevation varies exponentially with distance from the coast in order to give a geostrophic balance.

Kelvin waves move with the coast on their right in the Northern Hemisphere and on their left in the Southern Hemisphere. (From Mortimer, 1977.)

¹Wang, B. (2002) Kelvin Waves. Shankar, M., Ed., Elsevier Science Ltd., 7p. https://www.soest.hawaii.edu/MET/Faculty/bwang/bw/paper/wang_103.pdf

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