Analytical and numerical investigation of edge waves near a vertical breakwater over a convex bottom. (15th December 2022) - Zhang, Di; Wang, Gang; Feng, Xi; Feng, Weibing
Abstract: Vertical breakwaters are commonly built on the shore, and their existence must influence the character of nearshore dynamics. Analytical solutions based on linear-shallow water equation are derived for edge waves on a convex exponential profile which may be used to idealize the actual bottom profile with a vertical breakwater at the shoreline. The free surface elevation of the edge waves is described using the first kind of the Bessel functions of order ν . The amplitude profiles are similar to those of edge waves on a sloping beach but confined to a narrower offshore domain near the vertical breakwater. The dispersion relationship is derived from the no-flux boundary condition at the vertical breakwater. The wave number is not only related to the frequency but also the water depth at the vertical breakwater and the profile parameter. The group velocity increases rapidly to the maximum value with the increase of frequency, then decreases slowly and finally approaches the classic shallow water wave celerity of the depth at the vertical breakwater. The possible reason for this distinct behavior may be because dynamic characteristics of edge waves are highly dependent on the profile. Moreover, an extensively validated Boussinesq wave model is used to conduct numerical experiments for edge waves induced by surface bumps. The numerical results are consistent with the proposed solutions, confirming the validity of the new analytical solutions. Highlights: Analytical solutions of edge waves on a convex exponential profile are derived. The amplitude profiles are similar to those of edge waves on a sloping beach but confined to a narrower offshore domain. The group velocity increases rapidly to the maximum and then decreases slowly with the increase of frequency. Numerical experiments are conducted to validate the proposed analytical solutions. The study is hoped to improve the understanding of edge waves on a convex exponential profile.