On the existence of steady-state resonant waves in experiments

Abstract

This paper describes an experimental investigation of steady-state resonant waves. Several co-propagating short-crested wave trains are generated in a basin at the State Key Laboratory of Ocean Engineering (SKLOE) in Shanghai, and the wavefields are measured and analysed both along and normal to the direction of propagation. These steady-state resonant waves are first calculated theoretically under the exact resonance criterion with sufficiently high nonlinearity, and then are generated in the basin by means of the main wave components that contain at least 95 % of the wave energy. The steady-state wave spectra are quantitatively observed within the inherent system error of the basin and identified by means of a contrasting experiment. Both symmetrical and anti-symmetrical steady-state resonant waves are observed and the experimental and theoretical results show excellent agreement. These results offer the first experimental evidence of the existence of steady-state resonant waves with multiple solutions.

On the steady-state nearly resonant waves

“…deep water given by the HAM-based approach agreed well with the experimental observation in a laboratory, as mentioned by Liu et al (2015). In addition, unlike perturbation techniques, the HAM provides us great freedom to choose auxiliary linear operator L .…”

“…In particular, the multiple steady-state resonant waves were even observed experimentally (Liu et al 2015) in a basin at State Key Laboratory of Ocean Engineering (in Shanghai, China), with excellent agreement with their theoretical predictions given by means of the HAM.…”

“…For example, in case of k 2 /k 1 = 0.8915 (corresponding to a case of nearly resonant waves), the solution series converges (strictly speaking, here it is a kind of 'numerical' convergence, since we just numerically show the decrease of residual errors versus the order of approximation, which however is not a mathematical proof), rather quickly by means of −1.25 c 0 −0.75, with the optimal value of c 0 near −0.95, as shown in table 1. In fact, the numerical convergence of solution series given by the HAM for exactly resonant waves was generally guaranteed in the same way, as shown by Liao (2011), Xu et al (2012, Liu & Liao (2014) and Liu et al (2015). Note that the numerically convergent analytic approximations for the steady-state exactly resonant waves in values of the convergence-control parameter c 0 , with the corresponding distribution of wave energy 81.9 % (the first primary wave component), 9.7 % (the second primary one) and 8.2 % (the nearly resonant one).…”

“…In summary, our strategy is to transfer the nearly resonant wave problem (with two zero eigenvaluesλ 1,0 =λ 0,1 = 0 and one small eigenvalueλ 2,−1 that leads to small divisor problem) into the corresponding exactly resonant wave problem with three zero eigenvaluesλ 184 S. Liao, D. Xu and M. Stiassnie by means of choosing a generalized auxiliary linear operator (2.33) in the frame of the HAM. Note that the steady-state exactly resonant travelling waves in deep water and in finite depth governed by the full wave equations were successfully solved by means of the HAM (Liao 2011;Xu et al 2012;Liu & Liao 2014), and also observed experimentally in a laboratory with good agreement between analytic approximation and experimental results (Liu et al 2015). Here, it should be emphasized that, it is the HAM that provides us such kind of freedom to choose the generalized auxiliary linear operator (2.33), which is different from the linear part (2.23) of the original wave equations.…”

“…To the best of the authors' knowledge, this kind of steady-state nearly resonant wave has not been obtained analytically from the full wave equations. Therefore, considering the published results about the steady-state exactly resonant waves (Liao 2011;Xu et al 2012Xu et al , 2015Liu & Liao 2014;Liu et al 2015), steady-state gravity waves with time-independent amplitudes widely exist, no matter whether the resonance criterion is exactly or nearly satisfied.…”