The interaction of the tide with the outgoing current from a river can sometimes produce an unusual solitary wave that proceeds up the river from its mouth. Called a "tidal bore", this wave illustrates the depth dependance of the wave velocity of waves from the ocean.
The precondition for the creation of a tidal bore is the existence of an outflow current speed from the mouth of a river that exceeds the incoming speed of waves from the ocean associated with the incoming tide. The wavelength of tidal waves is longer than the depth of the shallow water near the shore, so they move shoreward with a velocity that depends upon the water depth. The limiting case expressions for the "celerity" or wave velocity with respect to the water are
From the shallow water velocity expression, you can see that the wave slows down as it approaches the shore in progressively shallower water. As it slows, the waves become steeper in the fashion of general ocean waves. In a shallow estuary, it is possible for the wave velocity to be slower than the outflow velocity from the river. This can effectively create a block of the incoming tide at the depth where the incoming wave velocity is equal to the river's outflow velocity. But the rising tide deepens the water and increases the wave velocity until it exceeds the river's outflow velocity. "Once this condition has been reached, the tidal crest sweeps into the estuary as a conspicuous wave or even as a frothy front - the tidal bore - following the passage of which the water levels inside and outside of the estuary remain more nearly equal." (von Arx)
The most famous of the tidal bores is that in the Tsientang Kiang estuary in China. "Here junks tie up securely against the rush of the bore, which on ocasion may be as much as 25 feet high, and schedule their trips to ride upstream on the current that follows."(von Arx) Tidal bores are more prevalent under the conditions of spring tides when the gravitational pulls of the Sun and Moon are in phase. They also occur in some straits such as Messina, the passage between Scylla and Charybdis. Tidal bores occur in the Amazon River, the Severn River in England, the Petitcodiac River at the head of the Bay of Fundy, and the Seine, Orne and Gironde Rivers of France.
Traveling Wave Concepts