The runup of long waves around piecewise linear bathymetries

The evolution of waves on beaches is the quintissential problem of coastal engineering. Most practical problems involve directional waveforms with complex spectral distributions. In the last ten years consensus has emerged that certain terminal effects such as coastal flooding and inundation are mainly affected by the infragravity waves, i.e. the long wave part of the incident spectrum. These waves can be described by the shallow-water wave equations, which are also the standard model for tsunamis or tidal waves. Interest in these equations has rekindled because comparisons with both large-scale laboratory data and field data have demonstrated a remarkable and surprising capability to model complex evolution phenomena, and in particular the maximum runup. The maximum runup is arguably the single most important parameter in the design of coastal structures such as seawalls and dikes and for evaluating the inundation potential of tsunamis.; A general method for solving developing exact solutions of the shallow-water wave equations is developed for determining the amplification factor of incident long waves as a function of the incident wave characteristics and the topographic variation. This method is then applied to the different ocean topographies composed of linearly varying depth segments and of constant depth segments, also known as composite beaches. Asysmptotic expressions are derived for the runup of solitary waves, and a series of large-scale laboratory experiments was conducted and is described. The analytical results are found in good agreement with the laboratory data for the time histories of free surface elevations and the maximum runup heights. An important result is that the maximum runup on the continental slope and shelf case is governed by the onshore slope, i.e., the slope which includes the initial shoreline, at least for the long waves of tsunami scales.; In the last part of the study the evolution of solitary waves around circular islands is examined both through a laboratory model and by developing analytical results. The time histories of free surface elevation at different locations and the maximum runup heights around the island were measured. Very good agreements between the analytical model and the experimental data was observed.; The solutions developed here are not only useful for developing solutions for most realistic initial conditions, but also useful for the preliminary design of coastal defense structures for tsunami attack, and for validating numerical codes which are used in the final design and for producing inundation maps, and whose robustness remains to be proven.