A study on nonlinear steady-state waves at resonance in water of finite depth by the amplitude-based homotopy analysis m
- PDF / 761,131 Bytes
- 12 Pages / 595.22 x 842 pts (A4) Page_size
- 7 Downloads / 141 Views
A study on nonlinear steady-state waves at resonance in water of finite depth by the amplitude-based Homotopy Analysis Method * Da-li Xu 1, Zeng Liu 2 1. College of Ocean Science and Engineering, Shanghai Maritime University, 1550 Haigang Ave, Shanghai 201306, China 2. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan 430074, China (Received March 3, 2019, Revised October 12, 2019, Accepted October 21, 2019, Published online March 25, 2020) ©China Ship Scientific Research Center 2020 Abstract: Nonlinear steady-state waves are obtained by the amplitude-based Homotopy Analysis Method (AHAM) when resonances among surface gravity waves are considered in water of finite depth. AHAM, newly proposed in this paper within the context of Homotopy Analysis Method (HAM) and well validated in various ways, is able to deal with nonlinear wave interactions. In waves with small propagation angles, it is confirmed that more components share the wave energy if the wave field has a greater steepness. However, in waves with larger propagation angles, it is newly found that wave energy may also concentrate in some specific components. In such wave fields, off-resonance detuning is also considered. Bifurcation and symmetrical properties are discovered in some wave fields. Our results may provide a deeper understanding on nonlinear wave interactions at resonance in water of finite depth. Key words: Nonlinear wave, wave resonance, steady state, homotopy analysis method
Introduction Resonant interactions among surface gravity waves are one of the physical processes governing the evolution of ocean wave spectra. It has been studied a lot since 1960, pioneered by Phillips' classic work[1]. The resonance conditions of gravity waves with two primary components are
uk1 vk 2 k u ,v
(1)
u1 v2 u ,v u ,v
(2)
where the angular frequency and vector wavenumber k fulfill the linear dispersion relation g | k | tanh(| k | d ) and d denotes water depth. u and v are integers. Exact ( u ,v 0 ) and near * Project supported by the National Nature Science Foundation of China (Grant Nos. 11602136, 51609090), the Science and Technology Commission of Shanghai Municipality (Grant No. 17040501600). Biography: Da-li Xu (1986-), Female, Ph.D., Lecturer Corresponding author: Da-li Xu, E-mail: [email protected]
( u ,v 0 ) resonances are represented by the
frequency detuning u ,v . Note that the quartet resonance of surface gravity waves is the lowest-order resonance in deep water and water of finite depth[1]. It is natural and necessary to start from the quartet resonance when we study the evolution of ocean wave spectra. Particularly, mainly for the sake of simplicity, the quartet resonance in the degenerate case (i.e. triads where one wave is counted twice) has been studied a lot since last century. In deep water, the resonant component was found to grow linearly with time in the initial stage[1-3] and then exchange wave energy with the primary ones in rel
Data Loading...
