A *tidal bore* is a sort of hydraulic jump that propagates up (i.e., upstream) a river estuary. The upper part of Figure 4.7 shows such a bore in the
local rest frame of the Earth. The bore is propagating at the velocity
up a river of uniform width, and depth
, that is flowing downstream at
the velocity
. The flow behind the bore is of depth
, and is flowing upstream at the velocity
. The lower part of the figure shows
the same phenomenon in the rest frame of the bore. In this frame, we observe a stationary hydraulic jump with an upstream depth and flow
velocity
and
, respectively, and a downstream depth and flow velocity
and
, respectively.
Making use of Equations (4.44) and (4.45), we obtain

(4.62) |

which can be rearranged to give

(4.63) |

Thus, we deduce that the speed of the bore, relative to the unperturbed river, is a simple function of the upstream and downstream depths. Note that, in the limit , the previous equation reduces to . In other words, a weak bore degenerates into an ordinary shallow water gravity wave propagating at the characteristic velocity relative to the stream.

Tidal bores are found in river estuaries where a funneling effect causes the speed of the incoming tide to increase to such a point that the flow becomes super-critical. For example, bores can be observed daily on the River Severn in England.