Oceanography

Solitary waves

Solitary waves are another class of non-linear waves, and they have very interesting properties. They propagate without change of shape, and two solutions can cross without interaction. The first solution was discovered by John Scott Russell (1808-1882), who followed a solitary wave generated by a boat in Edinburgh’s Union Canal in 1834.

The properties of a solitary waves result from an exact balance between dispersion which tends to spread the solitary wave into a train of waves, and non-linear effects which tend ot shorten and steepen the wave. The type of solitary wave in shallow water seen by Russell, has the form:

 Σ= a Sech ^h      3a

                        ( ----   ) ^{½                                             (x-ct)}

                          4d³

 

Which propagates at a speed:

 C= Co (1+  a   )

                    -

                   2d

You might think that all shallow-water waves are solutions because they are non-dispersive, and hence they ought to propagate without change in shape. Unfortunately, this is not true if the wave have finite amplitudes. The velocity of the wave depends on depth. If the wave consists of a single hump, then the water at the crest travels faster than water in the trough, and the wave steepens as it moves forward. Eventually, the wave becomes very steep and breaks. At this point it is called a bore. In some river mouths, the incoming tide is so high and the estuary so long and shallow that the tidal wave entering the estuary eventually steepens and breaks producing a bore that runs up the river.