**1. Introduction**

Ocean waves contain large untapped renewable energy sources, which could be used to fulfil the energy demand [1]. Wave energy has the characteristics of high power density and wide distribution, and is renewable [2]. Hitherto, about one thousand wave energy converter (WEC) inventions have been patented [3], and more than 200 of them have entered the model testing stage [2]. However, present research mainly focuses on promoting the conversion efficiency of wave energy converters by improved mechanical design [4]. There are few research results to improve the wave energy generation efficiency from the perspective of increasing the wave energy density of the target sea area. Taking the characteristics of Bragg resonance into account is a good idea for wave energy focusing.

Bragg reflection or resonance is initially referred to as a special physical optical phenomenon of X-rays [5]. In the 1980s, Davies [6] first studied that Bragg resonance occurs if the bottom wavelength is an integral multiple of half wavelength of the incident waves. A followed idea originated for coastal protection was put forward. The strong reflection of water waves by seabed bars was investigated experimentally [7,8] as well as theoretically [9–11]. Afterwards, Bailard et al. [12] have extensively studied theoretically, computationally and experimentally to learn explicitly the feasibility of using longshore seabed bars to protect the shore and summarized its effectiveness and limitations. Some issues, including the partially standing-wave pattern [13] and shoreward increase

**Citation:** Zhang, H.; Tao, A.; Tu, J.; Su, J.; Xie, S. The Focusing Waves Induced by Bragg Resonance with V-Shaped Undulating Bottom. *J. Mar.Sci. Eng.* **2021**, *9*, 708. https:// doi.org/10.3390/jmse9070708

Academic Editors: Shih-Chun Hsiao, Wen-Son Chiang and Wei-Bo Chen

Received: 19 May 2021 Accepted: 24 June 2021 Published: 27 June 2021

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of wave amplitude [14–16] by Bragg reflection, were exposed. Howard & Yu [17] and Weidman et al. [18] have demonstrated that a specific phase relationship of wave-bottom interactions is significant, and sometimes it can cause exponentially varying standing-wave patterns. It is illustrated that Bragg resonance leads to offward intense reflection, even shoreward wave amplitude increase. Considering the strong effect of Bragg resonance, more studies focus on coastal protection or beach erosion from the aspect of the backward of the seabed bars.

Subsequent investigations of Bragg resonance mechanism in the field of coastal protection and engineering application pay more attention to the actual situations, e.g., oblique incident waves [10,19–22], sub- and superharmonic frequency [19–23], and seabed configuration [10,21,22,24].

Recently, Couston et al. [22] have proposed to revamp the Bragg resonance mechanism as a means of coastal protection by considering oblique seabed bars that divert, rather than reflect, shore-normal incident waves to the shore-parallel direction. Indeed, the novelty that with two superposed sets of oblique seabed bars reflecting waves along the shore-parallel direction, becomes efficiently deflected far to the sides, leaving a wake of decreased wave activity downstream of the patch. However, the wave characteristics of the forward of the oblique seabed bars are not studied by Couston et al. By contrast, few studies concern the wave reflection of the forward. From another perspective of the seabed bars reflecting waves, the wave-bottom interactions due to Bragg resonance may be of interest and be more in accordance with wave power generation need.

In the cases where closed-form solutions cannot be obtained, a wide range of numerical models are available to study water-wave scattering by seabed topographies [22]. These include the extended versions, e.g., the mild-slope equations [11,25], the coupledmode approach [26,27], the integral matching/discretized bottom method [28,29], the fully nonlinear Boussinesq equation adjusted for rapid bottom undulations [30], an asymptotic linear analytical solution (ALAS) [24], and the high-order spectral method [31]. Numerical investigations of Bragg scattering have helped explain several discrepancies between theory and experiments, including the difference between the observed and predicted Class II Bragg resonance frequency due to evanescent modes [32] and the resonant frequency downshift/upshift for the subharmonic/superharmonic Class III Bragg condition due to high-order nonlinearity [30].

Simultaneously, Bragg resonance theory has made some achievements in application fields. A series of submerged breakwaters have been developed to achieve coastal protection based on the Bragg resonance characteristic of reflecting waves to the sea in recent years. For the two-dimensional numerical cases, the parameters' influence on the Bragg resonance reflection coefficient has been investigated by scholars from different aspects, e.g., the shape, height, width, the number of the submerged breakwaters, the slope of the seabed, and wave conditions [33–37]. Generally speaking, these numerical simulations are based on various theoretical methods to study normally incident waves over a series of rectangular or trapezoidal breakwaters. Ning et al. [38] investigated numerically and experimentally the effect of the breakwater shape on hydrodynamic behavior in a two-dimensional flume. Shih & Weng [39] studied the interactions between waves and various combinations of undulating breakwaters in a three-dimensional basin. The reflection coefficient, transmission coefficient, and attenuation of wave energy were analyzed. Liu et al. [40] examined the Bragg reflection of waves by multiple submerged semi-circular breakwaters, by considering obliquely and normally incident waves independently. Although two-dimensional flume or three-dimensional basin experimental studies have been carried out, the study based on Bragg resonance for improving wave energy generation has not been widely considered. At present, research on the use of wave energy is gradually carried out. Water flow and wave propagation can be altered by artificial terrains or control devices. Zheng et al. [41] carried out a laboratory study on wave-induced setup and wave-driven current in a 2DH reef-lagoon-channel system. Elandt et al. [42] found that gravity waves can focus at a specific location by a concave mirror or a convex

lens of gravity waves. Tao et al. [43] investigated experimentally strong wave reflection induced by Bragg resonance. They found that wave reflection is effective to amplify the free-surface oscillation amplitude and focus the wave energy in front of the undulating bottom. Zhang & Ning [44] confirmed that reflected waves from the parabolic opening of the breakwater can travel towards a fixed focus position. For a specified wave environment, the wave heights at the focus positions can reach over several times of the incident wave heights, indicating wave energy multiplication. They all proposed the concept of the focal point. Instead of putting a multitude of wave energy harvesting devices over a large area, a large wave energy absorber can be placed at the focal point. However, the influence of peculiar spatial bottom layouts is ignored, which also affect the focal point and focusing area. Previous physical experiment research [43] shows that waves are strongly reflected in front of the undulating bottom. Based on that, this study considers the wave-focusing effect in a specific area by optimizing the undulating bottom pattern.

The objective of this study is to investigate both the characteristics of Bragg resonance induced by the regular waves and random waves. Three main research questions need to be answered: (1) How does the angle of inclination for the V-shaped undulating bottom affect the wave-focusing characteristics induced by the regular waves? (2) What analytical methods are used to determine the optimal angle of the V-shaped undulating bottom? (3) How do the steepness and spectrum width of random waves affect the wave-focusing characteristics with the V-shaped undulating bottom, and what is the law of their influence on wave height? By the high-order spectral (HOS) numerical simulation, these questions have been examined. The concept of V-shaped undulating bottom is first proposed, i.e., a horizontal V-shaped pattern with two continuous undulating bottoms inclined at the same angle perpendicular to the shoreline. The V-shaped undulating bottom can not only play the role of reflecting waves in front of the undulations, but also exploit the advantage of symmetrical "V-shaped", which can focus waves on the central axis and increase the wave heights in the focusing areas. However, waves in the ocean are stochastic, and the random waves contain wave components of different frequencies, each of which has different Bragg resonance effect due to interactions with the undulating bottom of fixed frequency. Most of the previous studies on wave Bragg resonance only consider the regular waves at the dominant resonance frequency. In addition, there are few studies on Bragg resonance involving the interactions between random waves and undulating bottom. In this work, we have further studied the Bragg resonance characteristics of the interactions between random waves and V-shaped undulating bottom by the numerical simulation of HOS. This study has been organized as follows. In Section 2, the generalized Bragg conditions, the high-order spectral (HOS) method, and the model establishment are introduced. In Section 3, Bragg resonance reflection coefficients of regular waves are studied from the perspectives of different angles *θ* and different values *f*. In Section 4, the Bragg resonance focusing characteristics of regular waves and random waves for V-shaped undulating bottom are expounded. In Section 5, the conclusions are provided.
