**5. Conclusions**

In this study, the combined Bragg reflections of oblique water waves by periodic surface-piercing and submerged breakwaters were solved using the eigenfunction matching method (EMM). In the solution procedure, the surface-piercing structures and bottom profiles are sliced into a number of shelves separated by abrupt steps. The solution on each shelf is composed of eigenfunctions with unknown coefficients. Then, a system of linear equations is obtained by applying the conservations of mass and momentum. The present method was validated under three problems of oblique wave scattering by a rectangular surface-piercing structure over a flat bottom, behind a parabolic breakwater, and Bragg reflection by periodic rectangular/triangular surface-piercing structures over a flat bottom. Then, the proposed method was applied to solve the Bragg reflections of oblique water waves by periodic surface-piercing structures over periodic bottoms. For oblique Bragg reflections by periodic surface-piercing structures, numerical results indicated that the case with a larger incidence angle results in a more intensive Bragg reflection and the secondary resonance is stronger than the primary resonance. However, the results are completely opposite for oblique Bragg reflections by periodic submerged breakwaters. In addition, Bragg's law of oblique waves was applied in the numerical experiments. Some theoretical and physical studies on the resonance intensity and incidence angle are required in future investigation.

**Author Contributions:** Conceptualization, I.-F.T. and C.-C.T.; methodology, C.-C.T.; software, C.-C.T.; validation, C.-S.Y. and C.-C.T.; formal analysis, C.-S.Y.; investigation, C.-S.Y.; resources, I.-F.T. and C.-C.T.; data curation, C.-S.Y.; writing—original draft preparation, I.-F.T. and C.-S.Y.; writing—review and editing, I.-F.T. and C.-C.T.; *J. Mar. Sci. Eng.* **2020**, *8*, 522

visualization, C.-S.Y.; supervision, I.-F.T. and C.-C.T.; project administration, I.-F.T. and C.-C.T.; funding acquisition, I.-F.T. and C.-C.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** Ministry of Science and Technology, Taiwan: 107-2221-E-992-045-MY2.

**Acknowledgments:** The Ministry of Science and Technology of Taiwan is gratefully acknowledged for providing financial support to carry out the present work under the Grant No. MOST 107-2221-E-992-045-MY2.

**Conflicts of Interest:** The authors declare no conflicts of interest exist.
