**3. Results**

In this study, the three-dimensional Bragg resonance numerical model is validated from two aspects: the first one is to verify the regular wave reflection coefficients under different incident angles *θ*; the other one is to verify the regular wave reflection coefficients under different wavenumber ratios *f* (*f* is the ratio of 2 times the incident wavenumber (2*k*1) to the undulating bottom wavenumber (*k*b), i.e., *f* = 2*k*1/*k*b). This will prove the accuracy of the model in studying the focusing characteristics of Bragg resonance on V-shaped undulating bottom under different incident angles and different resonance wave numbers.

### *3.1. Bragg Resonance Reflection Coefficients of Regular Waves under Different Angles θ*

The Bragg resonance reflection coefficient is the most important parameter to measure the magnitude of Bragg resonance. The Bragg resonance reflection coefficient is calculated according to the method proposed by Liu and Yue [20]. The Bragg reflection resonance coefficients *R*(*θ*) of the Class I Bragg resonance at different incident angles *θ* are simulated by the three-dimensional HOS method, as shown in Table 2, which are compared with the perturbation theory solutions of Mei [19] (oblique incidence on a finite strip of bars) in Figure 2.

**Table 2.** Reflection coefficients *R*(*θ*) of Class I Bragg resonance at different incident angles *θ*.

**Figure 2.** Reflection coefficients *R*(*θ*) comparisons of Class I Bragg resonance.

Figure 2 shows that the three-dimensional HOS method results are in good agreemen<sup>t</sup> with the perturbation theory solutions of Mei. When *θ* = 0◦ (normal incidence), *R*(*θ*) is the largest and up to 0.72. As *θ* increases, *R*(*θ*) first decreases with an increasing decaying rate. When *θ* reaches 45◦, *R*(*θ*) decreases to 0, which means the wave propagation is not affected by the periodic undulating bottom. When *θ* > 45◦, *R*(*θ*) exhibits a second maximum point around *θ* = 64.5◦.

### *3.2. Bragg Resonance Reflection Coefficients of Regular Waves under Different Values f*

The wavenumber ratios *f* is determined as 0.9–1.1 due to the study of Liu and Yue [20]. For *f*, in the range of 0.9–1.1, 17 different wavenumber ratios *f* are uniformly selected to investigate the regular wave reflection coefficients with different values of *f* under the forward (*θ* = 0◦) and oblique (*θ* = 19.5◦) incident conditions. The results of the threedimensional HOS model and perturbation theory [19] are compared.

According to Figure 3, when *f* = 0.9–1.1, the Bragg resonance reflection coefficients increase first and then decrease with the increase of *f*. When *f* is slightly less than 1.0, the Bragg resonance reflection coefficients reach the peak, which also accords with the phenomenon of Bragg resonance frequency descending. The verification results confirm the reliability of the three-dimensional HOS numerical model.

**Figure 3.** Bragg resonance reflection coefficients of regular waves under different wavenumber ratios *f*: (**a**) Forward incident condition (*θ* = 0◦); (**b**) Oblique incident condition (*θ* = 19.5◦).
