Phase saddles and dislocations in two-dimensional waves such as the tides

Abstract

A two-dimensional scalar wavefield of fixed frequency contains, in general, points where the amplitude is zero and the phase is indeter­minate. On a map of contours of equal phase these wave dislocations (interference nulls) are accompanied by saddles. When an external parameter is changed dislocations can be created in pairs or a pair can meet and destroy one another. For the simplest single-frequency wave equation it is a topological necessity that two saddles should participate in this event; moreover, they have to lie, in the final stage before annihilation, on the circle whose diameter is the line joining the dislocations. Examples are given to show how this basic pattern is always ultimately attained even when initially the configuration is quite different. In tidal theory, where the dislocations are amphidromic points, the external parameter that moves them can be the frequency. An example of an annihilation event occurs in the South Atlantic, and a close pair of amphidromic points may explain anomalous tidal observations from the Antarctic Peninsula. The tidal current, as distinct from the tidal rise and fall, provides an example of a two or three-dimensional vector field, and it is pointed out that the singularities in this field are precisely the same as those to be found in the polarization field of an electromagnetic wave.