The manuscript titled “Asymptotics of long nonlinear coastal waves in basins with gentle shores” is authored by S. Yu. Dobrokhotov, D. S. Minenkov, and M. M. Votiakova. The affiliations of the authors are as follows: S. Yu. Dobrokhotov is affiliated with the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences and the Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences, both located in Moscow, Russia. D. S. Minenkov is also affiliated with the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences. M. M. Votiakova is affiliated with the Moscow Institute for Physics and Technology in Dolgoprudny, Moscow region, Russia.
This research paper addresses the scientific problem of modelling long nonlinear coastal waves in basins with gentle shores, focusing on the shallow water equations. The primary challenge investigated is the behaviour and properties of coastal waves, which are waves localized near shorelines, in basins with a smoothly varying depth. The study extends traditional linear wave models, like Stokes and Ursell waves, to incorporate nonlinear effects, providing a more comprehensive understanding of wave dynamics in real-world coastal environments.
The methods employed in this study involve constructing asymptotic solutions to the nonlinear system of shallow water equations. These solutions are represented in a parametric form and rely on a modified version of the Carrier-Greenspan transformation, which adapts the equations for a variable boundary domain. The authors derive these solutions by generating asymptotic eigenfunctions, referred to as quasimodes, of an operator that describes the wave dynamics in terms of a Hamiltonian system. This system considers the depth of the basin and the gravitational acceleration, forming a framework that can accommodate the nonlinear characteristics of the waves. The asymptotic solutions are specifically tailored to account for the localization of wave energy near the coastline, a critical feature in understanding coastal wave behaviour.
The main findings of the manuscript reveal that the asymptotic solutions describe a new class of long nonlinear coastal waves that are periodic or nearly periodic in time and are localized near the shorelines. These solutions generalize the concept of coastal waves for complex shorelines with variable slopes, demonstrating that such waves can be modelled effectively even in the presence of nonlinear effects. The study finds that the asymptotic eigenfunctions resemble the “whispering gallery” wave functions known in acoustics but are adapted to the context of coastal wave dynamics. Furthermore, the research shows that the wave trajectories, under the conditions studied, are normal to the boundary and that the traditional requirement of domain convexity is not necessary for the wave solutions to hold.
The significance of this research lies in its potential to improve the understanding of coastal wave phenomena, which has implications for coastal management and engineering. By extending the theoretical framework for coastal waves to include nonlinear effects, this study provides a more accurate tool for predicting wave behaviour in coastal regions with gentle slopes. This can lead to better predictions of wave run-up and the impact of waves on coastal structures and ecosystems, thereby contributing to more effective coastal protection and hazard mitigation strategies.
Overall, the manuscript makes a valuable contribution to the field of mathematical physics and coastal engineering by presenting a novel approach to modelling coastal waves that bridges the gap between linear and nonlinear wave theories. The thorough methodological development and the significant findings underscore the importance of considering nonlinear effects in coastal wave studies, paving the way for future research in this area.