*Article* **The Focusing Waves Induced by Bragg Resonance with V-Shaped Undulating Bottom**

**Haiming Zhang 1,2, Aifeng Tao 1,2,\*, Junhao Tu 1,2, Junwei Su 1,2 and Shuya Xie 1,2**


**\*** Correspondence: aftao@hhu.edu.cn; Tel.: +86-15295540618

**Abstract:** Intensive wave reflection occurs when the wavelengths of the incident waves and bottom undulations are in a 2:1 ratio. Existing studies have included the Bragg resonance phenomenon of waves passing over a continuous undulating bottom parallel to and oblique to the shoreline. More generally, the Bragg resonance mechanism is used as a means of coastal protection, rather than wave power generation. To focus the wave energy in a specific area, here, we propose sinusoidal sandbars of a horizontal V-shaped pattern, which is formed by two continuous undulating bottoms inclined at an angle to each other and the center axis of the angle is perpendicular to the shoreline. Based on the high-order spectral (HOS) numerical model, both the characteristics of Bragg resonance induced by the regular waves and random waves are investigated. In the scenario of regular waves, it shows that the wave-focusing effect is related to the angle of the V-shaped undulating bottom, and the optimal angle of inclination for the V-shaped undulating bottom is 162.24◦. On that basis, considering the interactions between the random waves and the V-shaped undulating bottom of 162.24◦, the Bragg resonance characteristics of random waves are studied. The *BFI* factor combining wave steepness and spectrum width can evaluate the focusing intensity of the Bragg resonance of the random waves. For *BFI*, in the range of 0.15–1.0, the values of *H*smax/*H*s0 linearly increase with the increase of *BFI*.

**Keywords:** three-dimensional Bragg resonance; regular waves; random waves; high-order spectral (HOS) method; Gaussian spectrum; V-shaped undulating bottom
