Wave Forces on a Partially Reflecting Wall by Oblique Bragg Scattering with Porous Breakwaters over Uneven Bottoms
Abstract
:1. Introduction
2. Materials and Methods
2.1. Problem Definition
2.2. Dispersion Relations and Eigenfunctions
2.3. Subaerial Porous Breakwaters
2.4. Eigenfunction Matching Method
2.5. Wave Force on the Partially Reflecting Wall
3. Results
3.1. A Rectangular Porous Structure near a Partially Reflecting Vertical Wall
3.2. Wave Force on the Vertical Wall
3.3. Multiple Porous Structures near a Totally Reflecting Vertical Wall
3.4. Trapezoidal Porous Breakwaters near a Porous Seawall
4. Discussion
4.1. Constructive Bragg Scattering by the Partially Reflecting Wall
4.2. Destructive Bragg Scattering by the Partially Reflecting Wall
4.3. Oblique Incidence
4.4. Periodic Porous Breakwaters
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
Nomenclature
coefficient of vertical eigenfunction | |
coefficient of vertical eigenfunction | |
coefficient of vertical eigenfunction | |
coefficient of vertical eigenfunction | |
incidence angle | |
porosity of porous media | |
wavelength of incident wave | |
angular frequency of incident wave | |
density of water | |
velocity potential on the shelf | |
surface elevation on the shelf | |
parameter for porous layer | |
phase angle | |
vertical eigenfunction | |
the first horizontal eigenfunction | |
the second horizontal eigenfunction | |
vertical eigenfunction for the larger total depth | |
vertical eigenfunction for the smaller total depth | |
three-dimensional gradient operator | |
three-dimensional Laplace operator | |
operator for porous pressure | |
amplitude of incident wave | |
amplitude of the half-cosine shaped breakwater in Section 4 | |
width of the rectangular porous breakwater in Section 3.1 and Section 3.2 or separation distance between half-cosine breakwaters in Section 4 | |
water depth on the shelf | |
friction coefficient of porous media | |
acceleration of gravity | |
total depth on the shelf | |
unit of complex numbers | |
transverse wavenumber | |
lateral wavenumber of the evanescent mode on the shelf | |
absolute wavenumber of the evanescent mode on the shelf | |
lateral wavenumber of the propagating mode on the shelf | |
absolute wavenumber of the propagating mode on the shelf | |
index of modes | |
index for shelves and steps | |
pressure on the shelf | |
index for constructive Bragg scattering | |
index for destructive Bragg scattering | |
time | |
inertial coefficient of porous media | |
fluid velocity or discharge velocity on the shelf | |
coordinate of the step | |
reference location of the | |
three-dimensional Cartesian coordinates | |
EMM unknown coefficients | |
EMM unknown coefficients | |
Separation distance between the last porous breakwater and the vertical wall in Section 3.1, Section 3.2 and Section 4. | |
or | variable for depth eigenfunction |
partially reflecting factor of the vertical wall | |
reflection coefficient | |
dimensionless horizontal wave force on the vertical wall | |
number of shelves plus one | |
number of evanescent modes | |
wave period of incident wave | |
wavelength of the periodic bottom in Section 4 |
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0 | 0.4167 | 0.4419 | 0.4387 |
1 | 0.3960 | 0.4070 | 0.4499 |
2 | 0.3962 | 0.4110 | 0.4555 |
3 | 0.3954 | 0.4085 | 0.4550 |
5 | 0.3953 | 0.4087 | 0.4560 |
10 | 0.3952 | 0.4088 | 0.4566 |
15 | 0.3952 | 0.4087 | 0.4566 |
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Chang, J.-Y.; Tsai, C.-C. Wave Forces on a Partially Reflecting Wall by Oblique Bragg Scattering with Porous Breakwaters over Uneven Bottoms. J. Mar. Sci. Eng. 2022, 10, 409. https://doi.org/10.3390/jmse10030409
Chang J-Y, Tsai C-C. Wave Forces on a Partially Reflecting Wall by Oblique Bragg Scattering with Porous Breakwaters over Uneven Bottoms. Journal of Marine Science and Engineering. 2022; 10(3):409. https://doi.org/10.3390/jmse10030409
Chicago/Turabian StyleChang, Jen-Yi, and Chia-Cheng Tsai. 2022. "Wave Forces on a Partially Reflecting Wall by Oblique Bragg Scattering with Porous Breakwaters over Uneven Bottoms" Journal of Marine Science and Engineering 10, no. 3: 409. https://doi.org/10.3390/jmse10030409