**4. Discussion**

After the model is validated, the proposed EMM model is applied to solve Bragg reflections of oblique water waves by periodic rectangular and triangular surface-piercing structures over periodic parabolic breakwaters as depicted in Figure 10. The parabolic breakwaters are defined by Equation (39) with *h*1 = 1 m and 2*c* = 0.5 m and the surface-piercing structures are either rectangular or triangular with 2*a* = 0.5 m and *d* = 0.25 m, as shown in Figure 8. In addition, *S* = 3 m is set such that the problem configuration is reduced to the previous example if *hb* = 0. The incidence angle α, structure or breakwater number *L*, and breakwater height *hb* are the parameters studied in the following.

As depicted in Figure 11, we extend the previous example by considering waves of different incidence angles α = 15*o* and α = 30*<sup>o</sup>*. In other words, we set *N* = 3 and *hb* = 0, and used 10 shelves to approximate each triangular structure. In the figure, Bragg's law is observed that intensive reflections occur for <sup>2</sup>*<sup>S</sup>*(cos α)/<sup>λ</sup> being equal to positive integers. Moreover, it is interesting to find that the case with a larger incidence angle results in a more intensive Bragg reflection. In addition, it is noticeable that the secondary resonance <sup>2</sup>*<sup>S</sup>*(cos α)/<sup>λ</sup> = 2 is stronger than the primary resonance <sup>2</sup>*<sup>S</sup>*(cos α)/<sup>λ</sup> = 1. In addition, total Bragg reflections occur in the secondary resonances.

**Figure 10.** Schematic diagram of Bragg reflections by periodic rectangular surface-piercing structures over periodic parabolic breakwaters.

**Figure 11.** The reflection coefficients with different incidence angles for Bragg reflections by periodic (**a**) rectangular and (**b**) triangular surface-piercing structures over flat bottoms.

Before studying the combined Bragg reflections of periodic surface-piercing and submerged breakwaters, problems of Bragg reflections solely by three periodic parabolic breakwaters of different heights are considered. Here, we use 40 shelves to approximate each parabolic breakwater as before and in the following. As depicted in Figure 12a, the case of higher breakwaters results in a more intensive Bragg reflection, as expected. In addition, Figure 12b shows the Bragg reflections for different incidence angles, which also confirm Bragg's law when <sup>2</sup>*<sup>S</sup>*(cos α)/<sup>λ</sup> is equal to a positive integer. However, it is interesting to observe that a larger incidence angle results in a less intensive Bragg reflection and the primary resonance <sup>2</sup>*<sup>S</sup>*(cos α)/<sup>λ</sup> = 1 is stronger than the secondary resonance <sup>2</sup>*<sup>S</sup>*(cos α)/<sup>λ</sup> = 2, which are totally opposite to the situations of Bragg reflections caused solely by periodic surface-piercing structures.

**Figure 12.** The reflection coefficients for Bragg reflections over periodic parabolic breakwaters of (**a**) different heights and (**b**) different incidence angles.

Then the combined Bragg reflections of periodic surface-piercing and submerged breakwaters are studied by setting different heights of the periodic parabolic breakwaters. As depicted in Figure 13, the EMM is applied to solve the problems of Bragg reflections of normal incident waves by three periodic rectangular surface-piercing structures over three periodic parabolic breakwaters. In the figure, it is clear that the case of higher breakwaters results in a stronger Bragg reflection, as expected. In addition, if the periodic breakwaters are high, total Bragg reflections occur for both the primary and secondary resonances.

**Figure 13.** The reflection coefficients for Bragg reflections by periodic rectangular surface-piercing structures over periodic parabolic breakwaters of different heights.

Then, the study with *hb* = 0.6 m in Figure 13 is extended to oblique incidence angles, as depicted in Figure 14a, which also numerically confirms Bragg's law of oblique waves. As depicted in Figure 14b, the results are similar if the study is extended to five periodic rectangular surface-piercing structures over five periodic parabolic breakwaters.

**Figure 14.** The reflection coefficients for Bragg reflections by periodic rectangular surface-piercing structures of different numbers (**a**) *L* = 3 and (**b**) *L* = 5 over periodic parabolic breakwaters.

Finally, if the surface-piercing structures are treated as floating rather than fixed, previous studies applied the EMM to problems involving a rectangular structure with surge [54,55], heave [56], and free motions [57]. Combing the proposed step approximation with these studies, the EMM can be applied to solve water-wave-scattering by a floating structure of different shapes over uneven bottoms. This is currently under investigation.
