On finite amplitude solitary waves-A review and new experimental data

Abstract

The existing analytical solutions for finite amplitude solitary waves, including the perturbation solutions, based on either the nonlinearity parameter, α = H / h, or the dispersion parameter, ϵ = k 2 h 2, and the closed form solutions, are reviewed. The convergence characteristics of the perturbation solutions are discussed, showing that the perturbation solutions for the velocity field diverge for large wave amplitude. The relationships between three existing closed form solutions are discussed. The analytical solutions are then compared with exact numerical solutions. The agreement is generally good for the free surface profiles, but not for the velocity field. One of the closed form solutions [Clamond, D. and Fructus, D., "Accurate simple approximation for the solitary wave,"C. R. Mec. 331, 727 (2003)] is in almost perfect agreement with the exact numerical solutions for both the free surface profiles and the velocity fields. New laboratory experiments, measuring both free surface profile and velocity field over a wide range of α values (up to 0.6) are then presented. High speed particle image velocimetry is used to measure the velocity field in the entire water column. Detailed comparisons among the experimental data, analytical theories, and numerical solutions show that for relatively small amplitude solitary waves, say, α ≤ 0.2, all theories and numerical results agree very well with the experimental data. However, when α ≥ 0.3 only [Clamond, D. and Fructus, D., "Accurate simple approximation for the solitary wave,"C. R. Mec. 331, 727 (2003)]'s solution and the numerical agree with the experimental data.