Ursell's edge waves are derived systematically using
a new method. Computed profiles
are then compared with the lesser known shoreline singular waves first
constructed by
Roseau (1958). A recent method of writing the continuous spectrum solutions
on a
plane beach is thereby extended to the discrete spectrum to enable the
reconstruction
of both types of edge waves so that, in particular, the unbounded wave
profiles are
more easily computed. The existence of stagnation points on the bed for
standing
edge waves is considered and demonstrated for the first few modes. A ramification
of this is the existence of (two-dimensional-cross-shore)
dividing ‘streamlines’ from
the bed to the surface also, the number of which appears to equate to the
modal
number of the edge wave. These dividing streamlines, along with other streamlines,
are computed for the first few modes of both the Ursell and the (alternative)
singular
waves constructed by Roseau.It follows that these waves can also exist in the presence of solid
cylinders bounded
by fixed streamlines and, in particular therefore, that the hitherto unbounded
Roseau
waves can exist in a bounded state since a region including the origin
can be removed
from the flow by exploiting a dividing streamline. It is confirmed that
the wavenumbers
of the Roseau waves interlace those of the Ursell waves. An examination
of available
evidence leaves open to further research the question of whether the alternative
Roseau waves have been ‘inadvertently’ observed either in the
laboratory or, by
means of contamination of data, in the field. Further laboratory simulations
using
longshore solid cylinders as ‘wave guides’ are proposed.
Rayleigh edge waves in two-dimensional crystals with Lorentz forces: From skyrmion crystals to gyroscopic media
