Reflection of a high-amplitude solitary wave at a vertical wall

Abstract

The collision of a solitary wave, travelling over a horizontal bed, with a vertical wall is investigated using a boundary-integral method to compute the potential fluid flow described by the Euler equations. We concentrate on reporting new results for that part of the motion when the wave is near the wall. The wall residence time, i.e. the time the wave crest remains attached to the wall, is introduced. It is shown that the wall residence time provides an unambiguous characterization of the phase shift incurred during reflection for waves of both small and large amplitude. Numerically computed attachment and detachment times and amplitudes are compared with asymptotic formulae developed using the perturbation results of Su & Mirie (1980). Other features of the flow, including the maximum run-up and the instantaneous wall force, are also presented. The numerically determined residence times are in good agreement with measurements taken from a cine film of solitary wave reflection experiments conducted by Maxworthy (1976).

Nodal DG-FEM solution of high-order Boussinesq-type equations

“…This effect is a pure nonlinear effect. As concluded by Cooker, Weidman and Bale (1997) on the basis of experimental data and in agreement with their numerical results, the phase-change during the reflection is spatially dependent and hence these waves are not solitons, which result in a (usually minor) loss of energy during the reflection of the solitary waves. Therefore the wave cannot maintain it's original shape after the reflection.…”

“…As experienced by both Cooker, Weidman and Bale (1997) and Madsen, Bingham and Liu (2002) it is difficult to model the thin jet shooting up numerically due to the rapid and subsequent collapse of the jet afterward, which demands sufficient spatial resolution in this region. However, we introduced a small amount of artificial viscosity in the vicinity of the wall in order to suppress spurious oscillations and therefore we are able to continue the detachment curve where previous results have not been obtained in Figure 7 a) even for the high-amplitude cases.…”

Head-on collision of two solitary waves and residual falling jet formation

“…They considered the reflection of a solitary wave by solving the Boussinesq equations analytically as well as numerically. Cooker et al (1997) using a Boundary Integral Equation Method (BIEM) for solving the fully nonlinear equations, showed that the wall residence time provides an unambiguous characterization of the phase shift incurred during the reflection. Bona and Chen (1998) as did Power and Chwang (1984) considered a Boussinesq system to study the head-on collision of solitary waves.…”

“…Note that Byatt-Smith (1971), Su and Mirie (1980), Mirie and Su (1982) and Bona and Chen (1998) considered the asymmetric case too. Among the fully nonlinear water wave equations, Cooker et al (1997) and Chan and Street (1970) considered waves with normalized amplitude a/ h up to 0.70. The latter authors were only concerned with the maximum run-up whereas Cooker and co-authors investigated more deeply the interaction.…”

Experimental Investigation on Reflection and Energy Loss of Tsunami Over the Permeable Beach

“…Cooker et al [17] have considered the reflection at a vertical wall of a solitary wave, using a boundary-integral numerical model and the images from a previous study have carried out by Maxworthy (1976). Silva et al [10] have presented a linear solution for the interaction of tsunamis with coastal defense structures.…”