Physics of Traveling Waves in Shallow Water Environment

Igor Shugan,Sergey Kuznetsov,Yana Saprykina,Yang-Yih Chen

doi:10.3390/w13212990

Abstract

We present a study of the physical characteristics of traveling waves at shallow and intermediate water depths. The main subject of study is to the influence of nonlinearity on the dispersion properties of waves, their limiting heights and steepness, the shape of solitary waves, etc. A fully nonlinear Serre–Green–Naghdi-type model, a classical weakly nonlinear Boussinesq model and fifth-order Stokes wave solutions were chosen as models for comparison. The analysis showed significant, if not critical, differences in the effect of nonlinearity on the properties of traveling waves for these models. A comparison with experiments was carried out on the basis of the results of a joint Russian–Taiwanese experiment, which was carried out in 2015 at the Tainan Hydraulic Laboratory, and on available experimental data. A comparison with the experimental results confirms the applicability of a completely nonlinear model for calculating traveling waves over the entire range of applicability of the model in contrast to the Boussinesq model, which shows contradictory and unrealistic wave properties for moderate wavelengths.

Highlights

Long water wave research has a history of more than two hundred years
We present the results of calculations using the improved SGN model [31] and fifth-order Stokes wave solutions for intermediate water depths [34,35]
We present the main characteristics of highly nonlinear traveling waves (FN) in comparison with classical Boussinesq (BM) solutions

Summary

Introduction

Long water wave research has a history of more than two hundred years. The high suitability of shallow water theories for various practical problems is still of unremitting interest. The two classical Korteweg and de Vries (KDV) [1] and Boussinesq models (BM) [2], together with the nonlinear Airy wave theory [3], laid the foundation for the study of waves in shallow water. Cnoidal traveling waves, which are solutions to the equations of KDV and BM, have been detailed and presented in an easy-to-use form by Wiegel et al [4,5]. The main physical characteristics of waves, such as period and wavelength, celerity, phase speed and other properties, were presented in visible and tabular form. An experimental study of cnoidal waves in shoaling water was presented in [6]

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Conclusion