Proceedings of the Thirty-second (2022) International Ocean and Polar Engineering Conference
Shanghai, China, June 5–10, 2022
Copyright © 2022 by the International Society of Offshore and Polar Engineers (ISOPE)
ISBN 978-1-880653-81-4; ISSN 1098-6189

www.isope.org

Numerical Study of Trapped Waves Propagation over a Submerged Step
Qian Wu,1,2, You Dong2, Xingya Feng1
1

Department of Ocean Science and Engineering, Southern University of Science and Technology, Shenzhen, China
2Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, China

deep-to-shallow area, the distribution of the wave falls to keep theoriginal equilibrium. Then the kurtosis on the depth transitions becomeslarger than 3, which shows a higher probability of the appearance of arogue wave. Thus, the distribution of wave statistics resulting from therapid water depth change is a fundamental interest for explaining therogue wave in the coastal region. The experiment of an irregular longcrested wave train travelling over a submerged trapezoidal bar wasconducted and notably increase of skewness and kurtosis of the wave
were also found in Trulsen et al. (2012). Zhang and Benoit (2021)
numerically studied the cases of one experimental setup from Trulsen etal. (2012) and further demonstrated the enhancement of kurtosis
triggered by the second and third-order effects. Viotti and Dias (2014)
verified the change of skewness and kurtosis of random waves
propagating over the depth transition by a spectral method.

ABSTRACTIn this paper, propagation of trapped waves traveling along a submerged
step is investigated numerically. Within the framework of the exact
model of surface gravity waves, a numerical model based on a conformal
mapping method is developed to simulate nonlinear wave propagation.
Regular incident waves over a range of driving frequencies and ratio ofthe depth in shallower water (h2) and deeper water (h1) are designed togenerate the first four trapped modes. For each of the trapped modes
considered, the time and spatial domain structure of the wave trapping
are obtained and analysed. The trapped harmonic wave amplitude ischaracterized by reflection coefficients and Fourier series coefficients, aswell as the parameters of skewness and kurtosis. The results of reflection
coefficients are found to agree well with the theoretical predictions. Inaddition, the linear and weakly nonlinear trapped waves induced by weak
resonant interactions of water waves are identified by the apparent
variation of the higher harmonic components. It is found that thenonlinearity grows with the increase of trapped modes and wave
steepnesses but has little effect on the reflection.

A much-debated question is the fully nonlinear and dispersion model of
the wave travelling over strong bottom discontinuity. By conducting
experiments and comparing with two theoretical models, Rey et al. (1992)
carefully investigated linear and weakly nonlinear gravity waves over a
submerged step, and the effects of sharp and bar corners on waves werecharacterized by the reflection coefficient and the higher-harmonic
waves. Zhang and Benoit (2021) proposed a fully nonlinear model witha harmonic extraction approach, and found that the de-shoaling leads tothe nonlinear and dispersive parameters. Meanwhile, Li et al.(2021; 2021)
developed a second-order theory and conducted experiments to study thenarrow-banded surface gravity wave packets traveling over a rectangularsubmerged bar, a super-harmonic wave released where the in-phasereflected wave packet propagated oppositely and the out-of-phasetransmitted packet travelled in the same direction of the main packet withslower speed. However, little research considers the resonancerelationship of the incident wave and the immersed step. There isevidence that the long wave trapped on the ridge can initiate rogue waves
in the coastal environment due to reflection and refraction effects.
Koshimura et al. (2001) successfully demonstrated the occurrence of the1996 Irian Jaya earthquake-tsunami by analyzing the incident wave angle
and the trapping efficiency. Similarly, in the latest research, Zheng et al.
(2016) derived the algebraic equations to describe the trapping
mechanism of a hyperbolic-cosine squared cross-sectional ridge where
the wave energy can be trapped on the ridge if a tsunami occurs.

KEY WORDS: Wave reflection; trapped mode; nonlinear waves;conformal mapping.

INTRODUCTION
The occurrence of abnormal high waves can bring coastal hazards.
Research on the cause of the extreme wave also termed ‘rogue’ or ‘freak’wave has attracted increasing interests (Madsen, Fuhrman, & Schäffer,2008; Ou, Liau, Hsu, & Tzang, 2002; Sheng, Alymov, & Paramygin,2010). As ocean waves propagate towards the near-shore region, theundulation bottom topography can affect them and lead to wave
refraction, wave reflection and shoaling (Lynett, Melby, & Kim, 2010;
Pelinovsky, Shurgalina, & Rodin, 2015). Various physical mechanism
has been proposed for the irregular wave dynamic process. One
mechanism is the nonlinear interaction of the surface wave and the strong
bottom discontinuity (Kharif & Pelinovsky, 2003).
Previous research has established that a great modification of wave
spectrum can be generated due to abrupt water depth change (Rey,Belzons, & Guazzelli, 1992; Claudio Viotti & Dias, 2014). To
characterize the occurrence of the extreme wave, the parameter kurtosis(the degree of tails of a normal distribution) is used, where the value ofkurtosis is 3 for a Gaussian process. When the wave propagates over aThis paper attempts to investigate the nonlinear mechanism of the wave
propagating over an intermediate-to-shallow bathymetry by numericalcomputations of a nonlinear model using conformal mapping method.

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