**4. Discussion**

*4.1. Bragg Resonance Focusing Characteristics of Regular Waves for V-Shaped Undulating Bottom*

In the section, the wavelength of the incident wave is 1 m, which need 16 nodes to represent. The other parameters of the incident waves and undulating bottom can be calculated according to the section of the Model Establishment.

### 4.1.1. Spatial Distributions of Wave Amplitudes in the Focusing Areas for Regular Waves

The V-shaped layout can be regarded as the combination of two parts of oblique bottom undulations relative to the incident waves. The wave-focusing effects, therefore, are the effects of two parts of oblique bottom undulations relative to the incident waves. When the wave incident angle (between *k*1 and *k*b) *θ* is greater than 45◦, *R*(*θ*) has a secondary maximum. There is a relationship between *α* and *θ*, as shown in Figure 1.

As *θ* increases, the *α* of the V-shaped layout becomes smaller. As a result, the wavefocusing area becomes smaller, so the wave-focusing effect is weakened. The values of *α* parameter for the specific simulation cases are listed in Table 1.

Bragg resonance occurs when regular waves pass over the V-shaped undulating bottom and satisfy the conditions that the wavelengths of the incident waves and bottom undulations are in a 2:1 ratio. Incident waves and reflected waves from two directions superimposed in front of the bottom causes wave energy focusing effect. Under the condition of regular wave incidence with the same wave amplitude, the Bragg resonance effects of wave-bottom interactions at different V-shaped undulating bottom angles *α* are simulated by the three-dimensional HOS numerical model. The maximum wave amplitudes calculated (*A*max) are compared with the incident wave amplitude (*A*0), and the spatial distribution characteristics of wave amplitudes are analyzed by *A*max/*A*0. The maximum wave amplitudes calculated (*A*max) are different at different node positions. So, the values of *A*max/*A*0 is also different. Table 3 shows that the calculated wave amplitude increases change at the focal points corresponding to different *α* (90◦ ≤ *α* ≤ 180◦). The node position of the maximum value of *A*max/*A*0 for each simulation case is called the focal point, and then *<sup>A</sup>*maxp/*A*<sup>0</sup> is used to denote the value of *A*max/*A*0 at the focal point. When *α* = 180◦ (the incident wave direction *k*1 is parallel to the bottom *k*b), the calculated wave amplitude at the focal point increases to 1.81 times the initial incident wave amplitude. As *θ* increases, the *α* of the V-shaped layout becomes smaller. As a result, the wave-focusing area becomes smaller, so the wave-focusing effect is weakened. In conclusion, 90◦ ≤ *α* ≤ 180◦ is mainly considered in this study.


**Table 3.** The calculated wave amplitude increases at the focal points corresponding to different angles *α*.

To better analyze the spatial distribution characteristics of wave amplitudes, *α* = 178.21◦, 168.40◦, 162.24◦, 155.32◦, 148.58◦, and 140.47◦ are selected for comparison. The spatial distributions of wave amplitudes in the focusing areas due to Bragg resonance at different V-shaped undulating bottom angles *α* are analyzed, the values of *A*max/*A*0 are shown in Figure 4.

**Figure 4.** The values of *A*max/*A*0 in the scenarios of regular waves under different angles α of V-shaped undulating bottom (The dashed outline refers to the V-shaped undulating bottom range).

Figure 4 shows that no matter what degree *α* is, there are obvious wave-focusing areas and a series of focal points in front of the V-shaped undulating bottom. According to the principle of coastal protection, the wave amplitudes behind the bottom are possibly weakened owning to enormous waves reflected in front of the bottom. The V-shaped undulating bottom is generally symmetrical about the central axis; the wave-focusing areas are also symmetrical about the central axis. In addition, the wave-focusing areas are generally expand from right to left in front of the bottom in a "V" shape that is slightly smaller than the "V" shape of the bottom.

Figure 4 also shows that through the 6 selected representative *α* angles of the V-shaped undulating bottom, the spatial distributions of the wave amplitudes in the focusing areas owning to Bragg resonance are symmetric about the center of the V-shaped undulating bottom, and the largest wave amplitudes are along the axis of symmetry. By comparison, it is found that the wave-focusing intensity first increases and then weakens with the decreasing V-shaped undulating bottom angles *α* = 178.21◦, 168.40◦, 162.24◦, 155.32◦, 148.58◦, and 140.47◦. When *α* is about 162.24◦, there are significant focusing areas and focal points, and the focusing intensity at this angle is the strongest.

### 4.1.2. Quantitative Analysis of Wave Amplitudes in the Focusing Areas for Regular Waves

To further quantitatively analyze and compare wave amplitude focusing characteristics at different angles *α*, the wave amplitude increases *A*max*/A*<sup>0</sup> are classified and counted. In addition, under the four levels of *A*max*/A*<sup>0</sup> > 1, *A*max*/A*<sup>0</sup> ≥ 1.5, *A*max*/A*<sup>0</sup> ≥ 2, and *A*max*/A*<sup>0</sup> ≥ 2.5, the occurrence frequencies of the values of *A*max*/A*<sup>0</sup> in the whole simulation range are calculated and expressed respectively in the form of *P*(*A*max*/A*<sup>0</sup> > 1), *P*(*A*max*/A*<sup>0</sup> ≥ 1.5), *P*(*A*max*/A*<sup>0</sup> ≥ 2) and *P*(*A*max*/A*<sup>0</sup> ≥ 2.5), respectively. The frequency formula of each level is as follows:

$$P(A\_{\text{max}} / A\_0 \ge \mathbb{C}) = \frac{N(A\_{\text{max}} / A\_0 \ge \mathbb{C})}{N\_x \times N\_y} \tag{5}$$

where *C* is the specified level constant; *N* (*A*max*/A*<sup>0</sup> ≥ *C*) is the number of nodes corresponding to *A*max*/A*<sup>0</sup> ≥ *C*; *Nx* and *Ny* are the numbers of nodes on the *x* and *y*-axis in the whole simulation range respectively. In this section, *C* is assigned 1, 1.5, 2, and 2.5 respectively, *Nx* = *Ny* = 512.

For example, *α* = 178.21◦, 176.42◦, 174.63◦, 168.40◦, 167.52◦, 162.24◦, 160.50◦, 155.32◦, 151.93◦, 148.58◦, and 140.47◦ are selected for comparison. As there is no wave-focusing effect when *α* = 90◦, it is not analyzed. Figure 5 shows the level-frequency statistical diagrams of wave amplitude changes at different angles *α*.

In Figure 5a, it shows that a series of nodes appear in the simulation range at all selected angles, but *P*(*A*max*/A*<sup>0</sup> > 1) and *P*(*A*max*/A*<sup>0</sup> ≥ 1.5) gradually decrease with the decrease of angles *α*. *P*(*A*max*/A*<sup>0</sup> > 1) decreases from 21.49% to 15.74%, while *P*(*A*max*/A*<sup>0</sup> ≥ 1.5) decreases from 5.32% to 0. It can be seen that the focusing area is decreasing. That is because the area of Bragg resonance decreases with the decrease of the angle *α*, and then the wave height focusing areas also reduce. In addition, that is why the range of angles *α* of V-shaped undulating bottom is 90◦ ≤ *α* ≤ 180◦.

However, the wave-focusing effects are not only judged by the wave height in the focusing areas but also based on the overall amplitude increases of nodes in the wave height focusing areas, i.e., the values of *A*max*/A*0. Therefore, Figure 5b further shows the frequency statistic results of *P*(*A*max*/A*<sup>0</sup> ≥ 2) and *P*(*A*max*/A*<sup>0</sup> ≥ 2.5). It shows that both *P*(*A*max*/A*<sup>0</sup> ≥ 2) and *P*(*A*max*/A*<sup>0</sup> ≥ 2.5) increase first and then decrease with the decrease of angles *α*. Moreover, when 160◦ < *α* < 168◦, the overall value of *P*(*A*max*/A*<sup>0</sup> ≥ 2) is larger. That is to say, the total number of nodes of *A*max*/A*<sup>0</sup> ≥ 2 is the largest. Meanwhile, when *α* = 162.24◦, the values of *P*(*A*max*/A*<sup>0</sup> ≥ 2) and *P*(*A*max*/A*<sup>0</sup> ≥ 2.5) are the largest in the simulation range, which implies that the number of nodes corresponding to *A*max*/A*<sup>0</sup> ≥ 2 and *A*max*/A*<sup>0</sup> ≥ 2.5 is the most, and the wave-focusing effect is the strongest. The spatial distribution of wave amplitude change shows that when *α* = 162.24◦, the scope of wave height focusing areas are relatively concentrated, the wave amplitude increases are the largest as a whole, and the wave-focusing effect is the best.

In conclusion, *α* = 162.24◦ is considered to be the optimal angle for V-shaped undulating bottom.

**Figure 5.** Level-frequency statistical diagrams of wave amplitude changes under different angles *α*: (**a**) *A*max*/A*<sup>0</sup> > 1 and *A*max*/A*<sup>0</sup> ≥ 1.5; (**b**) *A*max*/A*<sup>0</sup> ≥ 2 and *A*max*/A*<sup>0</sup> ≥ 2.5.

*4.2. Bragg Resonance Focusing Characteristics of Random Waves for V-Shaped Undulating Bottom* 4.2.1.TheGaussianSpectrum

The Gaussian spectrum is a symmetric spectrum, and its spectrum pattern is simple and regular. The expression of the Gaussian spectrum requires only 3 parameters, compared with the Jonswap formula, which requires 5 parameters. Therefore, it is more convenient to use the Gaussian spectrum to calculate. In this paper, the two-dimensional Gaussian spectrum pattern is selected, and the expression is shown in Equation (6).

$$S(k\_i) = \frac{\eta^2}{\sqrt{2\pi}\sigma} \exp\left(-\frac{\left(k\_i - k\_0\right)^2}{2\sigma^2}\right) \tag{6}$$

where: Gaussian spectrum expression *<sup>S</sup>*(*ki*), the standard deviation of the height of the wave surface *η* = ( *S*(*k*)*dk*)1/2 = √*<sup>m</sup>*0, zero-order spectrum moment *m*0, spectral peak wave number *k*0, spectral width parameter *σ*, value range of the *i*-th wave number 0 ≤ *ki*/*dk* ≤ *N*/4, the node number in the simulation range *N*.

The initial wave steepness and spectral width of the random wave are changed by controlling *η* and *σ* respectively. The initial wave steepness *ε*0, initial amplitude *a*0, and spectral width *B*0 are calculated according to Equations (7)–(9) respectively.

$$
\varepsilon\_0 = 2\eta k\_0 \tag{7}
$$

$$a\_0 = \varepsilon\_0 / k\_0 = 2\eta = 2\sqrt{m\_0} \tag{8}$$

$$B\_0 = \sigma / k\_0 \tag{9}$$

The amplitude of the *i*-th wave can be calculated by Equation (10).

$$a\_i = \sqrt{2S(k\_i)dk} \tag{10}$$

The initial wave surface and potential function can be obtained from Equation (11).

$$\begin{array}{l} \eta(\mathbf{x},0) = \sum a\_i \cos(k\_i \mathbf{x} + \theta\_i) \\ \phi^s(\mathbf{x},0) = \sum \frac{\mathbf{g}\_{i\cdot}^{ai}}{\omega\_i} \sin(k\_i \mathbf{x} + \theta\_i) \end{array} \tag{11}$$

where: the initial wave surface *η*(*<sup>x</sup>*, <sup>0</sup>), the initial potential function *φ<sup>s</sup>*(*<sup>x</sup>*, <sup>0</sup>), the initial phase of the *i*-th wave generated randomly through the program *θi*, the gravitational acceleration *g*, the circular frequency of the *i*-th wave *ωi*, according to the dispersion relation *ω*<sup>2</sup> = *gk*tanh*kh* (water depth *h*) in the finite water depth to calculate the *ki*.

Assuming that the initial wave surface and the initial potential function along the *y*-axis are equal, the two-dimension random wave field is extended into a three-dimension random wave field along the *y*-axis.

### 4.2.2. Evolution Characteristics of Random Waves on Flat Bottom

The Gaussian spectrum is used to generate the initial wave field. The initial phases of each simulated wave field are randomly generated. Different initial phases, dispersion relation, and wave modulation instability will cause uneven wave height distributions along the *x*-axis. Therefore, the evolution characteristics of random waves on the flat bottom are first studied in this study. The distributions of significant wave heights *H*s along the *x*-axis under three random initial phases in the wave order *M* = 3 are shown in Figure 6. It shows that the distributions of significant wave heights along the *x*-axis are greatly different with different initial phases. Here, the node number along the *x*-axis is 512, and 16 nodes represent one wavelength of incident free surface.

As the number of simulation increases, the significant wave heights *H*s under different random initial phases are averaged. The results are shown in Figure 7.

With the increase of the simulation groups, the distributions of significant wave heights *H*s along the *x*-axis become more uniform. When the number of simulation groups is more than 10, the significant wave heights are basically stable in the form of a horizontal line, which meets the requirements of analysis.

**Figure 6.** Distributions of significant wave heights *H*s along the *x*-axis under three random initial phases for *M* = 3: (**a**) Initial phase one; (**b**) Initial phase two; (**c**) Initial phase three.

**Figure 7.** Distributions of average significant wave heights *H*s along the *x*-axis under five and ten simulations for *M* = 3.

### *4.3. Evolution Characteristics of Random Waves on V-Shaped Undulating Bottom*

*BFI* was proposed by Janssen [56], which is determined by wave steepness and spectrum width. The initial definition is shown in Equation (12), which is converted into the definition of wavenumber spectrum (14) by combining with Equation (13):

$$BFI = \sqrt{2}\varepsilon\_0 / (2\Delta\omega/\omega\_0) \tag{12}$$

$$2\Delta\omega/\omega\_0 = \Delta k/k\_0\tag{13}$$

$$BFI = \sqrt{2} \varepsilon\_0 / B\_0 \tag{14}$$

where: the circular frequency *ω*, the initial circular frequency *ω*0, the wavenumber of incident random waves *k*, the wavenumber of initial incident random waves *k*0, the spectrum width *B*, the initial spectrum width *B*0, the initial wave steepness *ε*0.

Janssen proposed that when *BFI* is greater than or equal to 1.0, satisfying the conditions for generation of modulation instability, the possibility of freak waves is increased. To eliminate the influence of wave modulation instability on the simulation results, the *BFI* is set at less than 1.0 in this study. Under different combinations of initial wave steepness and spectrum width, the values of *BFI* are listed in Table 4.


**Table 4.** The values of *BFI* under the settings of the Gaussian spectrum.

The calculated wave heights (*H*smax) are compared with the incident wave height (*H*s0). To examine the interactions between random waves and undulating bottom, the spatial distribution characteristics of wave heights due to Bragg resonance are analyzed. Analyzing the combinations of initial wave steepness and initial spectrum width, it is found that there is a good linear relationship between *H*smax/*H*s0 and *BFI*, as shown in Figure 8. For *BFI*, in the range of 0.15–1.0, the values of *H*smax/*H*s0 linearly increase with the increase of *BFI*, so the fitting formula is shown in Equation (15):

*H*smax/*H*s0 = 0.39 × *BFI* + 1.09

**Figure 8.** The changing curve of wave height ratio *H*smax/*H*s0 at the focal points as *BFI* changes.

The goodness of fit (*R*2) in Equation (15) can reach 0.94, indicating that the formula fits well. It is speculated that the calculation results will be more consistent with the line with the increase of the simulation groups. When the value of *BFI* is more than 0.4, there is a linear relationship between *H*smax/*H*s0 and *BFI*; whereas the value of *BFI* is less than 0.4,*H*smax/*H*s0 and *BFI* are weakly linear relation. When the *BFI* factor is small, in other words,the initial wave steepness of a given spectrum width is small, which indicates that the nonlinear wave interaction is weak. However, a given initial wave steepness has a larger spectrum width, indicating that the frequency bandwidth of wave height distribution is larger. In summary, the smaller the *BFI* factor is, the smaller *H*smax/*H*s0 is; *BFI* and *H*smax/*H*s0 are becoming more independent, when the incident wave nonlinearity is low.
