**3. Results**

The present model is validated using three numerical examples.

#### *3.1. Rectangular Surface-Piercing Structure over a Flat Bottom*

First, we consider monochromatic wave trains with incidence angles α = 45o and α = 75o, which propagate towards a rectangular surface-piercing structure over a flat bottom with water depth *h* = 1 m, as depicted in Figure 3. The breadth and depth of the rectangular barrier are set to 2*a* = 0.6 m and *d* = 0.2 m, respectively. Figure 4 shows a comparison of the reflection and transmission coefficients obtained by the proposed EMM with the results obtained by Lebreton and Margnac [53], Bai [43], Söylemez and Gören [45]. In the figure, the convergence with respect to the increasing numbers of evanescent modes, *N*, is obvious. Both the reflection and transmission coefficients evaluated by the present model with *N* = 5 are in good agreemen<sup>t</sup> with those in the literature for the whole

frequency range. This validates the present model for solving problems of oblique wave scattering by a rectangular surface-piercing structure over a flat bottom.

**Figure 3.** Schematic representation of the water wave scattering by a rectangular surface-piercing structure over flat bottom.

**Figure 4.** Comparison of the reflection and transmission coefficients from the present study with the results in the literatures for water wave scattering by a rectangular surface-piercing structure over flat bottom with incidence angles (**a**) α = 45o and (**b**) α = 75o.

#### *3.2. Rectangular Surface-Piercing Structure behind Parabolic Breakwater*

We now consider oblique monochromatic wave trains that propagate towards a rectangular surface-piercing structure behind a parabolic breakwater defined by *z* = −*h*(*x*) for |*x*| ≤ *c* as

$$h(\mathbf{x}) = (h - h\_b) \Big( 1 + \frac{h\_b \mathbf{x}^2}{(h - h\_b)c^2} \Big). \tag{39}$$

As shown in Figure 5, the other parameters are set as *h*1 = 30 m, *hb* = 15 m, *d* = 7.5 m, 2*c* = 200 m, *w* = 20 m, and 2*a* = 30 m, which are exactly the same values as those of Manisha et al. [47]. Furthermore, 40 shelves are used to approximate the parabolic breakwater after performing a preliminary convergence analysis as shown in Figure 6.

**Figure 5.** Schematic diagram of the water-wave-scattering by a rectangular surface-piercing structure behind a parabolic breakwater.

**Figure 6.** Convergence analysis of *M* for the water wave scattering by a rectangular structure behind a parabolic breakwater.

Figure 7 shows the comparison of reflection coefficients obtained by the present method and those from Manisha et al. [47]. In the figure, the convergence with respect to the increasing numbers of evanescent modes, *N*, can also be observed. The convergen<sup>t</sup> results of the present model are in good agreemen<sup>t</sup> with those in Manisha et al. [47]. This validates the proposed EMM for solving oblique wave scattering by a rectangular surface-piercing structure over uneven bottoms.

**Figure 7.** Comparison of the reflection coefficients from the present study with the results from the literature for the (**a**) normal and (**b**) oblique water-wave-scattering by a rectangular structure behind a parabolic breakwater.

#### *3.3. Bragg Reflections by Periodic Surface-Piercing Structures over Flat Bottom*

As the final example of validation, we consider normal monochromatic wave trains, which propagate towards a series of periodic rectangular and triangular surface-piercing structures over flat bottoms with water depth *h* = 1 m. As depicted in Figure 8, the other parameters are set as α = 0, *L* = 3, *d*/*h* = 0.25, *a*/*h* = 0.25, and *S*/*h* = 3, which are exactly the same values as those of Ding et al. [10]. Typically, 10 shelves are adopted to approximate each triangular structure in this example, as shown in Figure 9b.

**Figure 8.** Schematic diagram of Bragg reflections by periodic (**a**) rectangular and (**b**) triangular surface-piercing structures over flat bottom.

**Figure 9.** Comparison of the reflection coefficients from the present study with the results in the literature for Bragg reflections by periodic (**a**) rectangular and (**b**) triangular surface-piercing structures over flat bottom.

Figure 9 shows the comparison of the reflection coefficients obtained by the present method and Ding et al. [10]. In the figure, the convergence with respect to the increasing numbers of evanescent modes, *N*, is also obvious, and the results by the present model with *N* = 3 are in good agreemen<sup>t</sup> with those obtained by Ding et al. [10]. In addition, Bragg's law confirms that intensive reflections occur for 2*S*/λ, being positive integers. This validates the present model for solving Bragg reflections of normal water waves by periodic surface-piercing structures of arbitrary shapes over flat bottoms.
