Standing edge waves reinvestigated: A comparison between two different bathymetries

As a result of wave refraction caused by variable water depth within enclosed or semi-enclosed nearshore areas like harbors and bays, standing edge waves play an important part in the circulation patterns. The bathymetries of these areas may fall into two categories as follows: one is a reflective beach with a moving shoreline where the waves run up and down and the other has a certain water depth at a fixed backwall. A comparison of the standing edge waves trapped on these two types of bathymetries is made. Analytical investigations show that the trapped modes may behave dissimilarly on each type of bathymetry, especially for relatively high modes when the bathymetry is not simply a constantly sloping beach but a piecewise one. Wave patterns induced by water surface disturbances of the numerical simulations are analyzed with wavelet spectra. Frequencies of different components of the standing edge waves are compared with theoretical predictions. The results of the bathymetry with a reflective backwall are consistent with the findings of previous studies. For the case with a moving shoreline, several very low modes of the standing edge waves can survive and persist into a steady state, whereas higher modes may suffer from a quick attenuation. The occurrence of the trapped modes is revealed sensitive to the initial position of the water surface agitations in this case.

A note on tuning in Roseau's alternative edge waves

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.

Mechanisms for the generation of edge waves over a sloping beach

Two mechanisms for the generation of standing edge waves over a sloping beach are described using classical linear water-wave theory. The first is an extension of the result of Yih (1984) to a class of localized bottom protrusions on a sloping beach in the presence of a longshore current. The second is a class of longshore surface-pressure distributions over a beach. In both cases it is shown that Ursell-type standing edge-wave modes can be generated in an appropriate frame of reference. Typical curves of the mode shapes are presented and it is shown how in certain circumstances the dominant mode is not the lowest.

Finite amplitude effects in free and forced edge waves

The effect of non-linearity on standing edge waves is studied on the basis of shallow water theory. Four problems are considered: the decay of free edge waves and the forcing of edge waves by an incident wave of double the frequency, a synchronous incident wave and by a side-wall wavemaker. Hysteresis effects are predicted for all types of forcing.

On the excitation of edge waves on beaches

The excitation of standing edge waves of frequency ½ω by a normally incident wave train of frequency ω has been discussed previously (Guza & Davis 1974; Guza & Inman 1975; Guza & Bowen 1976) on the basis of shallow-water theory. Here the problem is formulated in the full water-wave theory without making the shallow-water approximation and solved for beach angles β = π/2N, where N is an integer. The work confirms the shallow-water results in the limit N [Gt ] 1, shows the effect of larger beach angles and allows a more complete discussion of some aspects of the problem.