The series solution to the modified mild-slope equation for wave scattering by Homma islands

Wave Motion ◽  
2013 ◽  
Vol 50 (4) ◽  
pp. 869-884 ◽  
Author(s):  
Huan-Wen Liu ◽  
Jian-Jian Xie
Keyword(s):  
Wave Scattering ◽  
Series Solution ◽  
Mild Slope Equation

A Series Solution for Wave Scattering by a Circular Island on a Shoal Based on the Mild-Slope Equation

2012 ◽  
Vol 28 (1) ◽  
pp. 41-51 ◽  
Author(s):  
Y.-M. Cheng ◽  
C.-T. Chen ◽  
L.-F. Tu ◽  
J.-F. Lee
Keyword(s):  
Analytical Solution ◽  
Wave Scattering ◽  
Present Method ◽  
Series Solution ◽  
Shallow Water Equation ◽  
Variable Coefficients ◽  
Fourier Cosine ◽  
Mild Slope Equation ◽  
Scattering Field ◽  
Run Up

ABSTRACTA series solution based on the mild-slope equation is produced in this study of wave scattering produced by a circular cylindrical island mounted on an axi-symmetrical shoal. The solution is presumed to be a Fourier cosine expansion with variable coefficients in the radial direction on account of the symmetric scattering field, which translates the original 2-D boundary-value problem to a 1-D one in which an ordinary differential equation is in effect treated. Approximations to the coefficients of the governing equation with the Taylor expansions enable the use of the Frobenius method, and consequently the solution is obtained in a combined Fourier and power series. For verification, the present method is mainly compared with Zhu and Zhang's [1] analytical solution of the linearised shallow water equation for a conical shoal, and with a different analytical solution of the mild-slope equation developed by Liu et al. [2] for a paraboloidal shoal. Fine agreements are achieved. The present method is then used to investigate the variation pattern of the wave run-up when the shoal profile varies from conical to paraboloidal, and some interesting phenomena are observed.

A new model for the three-dimensional propagation of water waves in the presence of vertically sheared, horizontally varying currents

2020 ◽  
Author(s):  
Julien Touboul ◽  
Kostas Belibassakis
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Wave Scattering ◽  
Three Dimensional ◽  
Water Wave Scattering ◽  
Coupled Equations ◽  
Coupled Mode ◽  
Constant Shear ◽  
Mild Slope Equation ◽  
Sea Bed

<p>In coastal areas, steep bathymetries and strong currents are often observed. Among several causes, the presence of cliffs, rocky beds, or human structures may cause strong variations of the sea bed, while oceanic circulation, tides, wind action or wave breaking can be responsible for the generation of strong currents. For both coastal safety and engineering purposes, there are many interests in providing efficient models predicting the nonlinear, phase resolved behavior of water waves in such areas. The difficulty is known to be important, and many models achieving that goal are described in the related literature.</p><p>Recently, it was established that beneath the influence of vertically uniform currents, the vorticity involved in depth varying mean flows could have significant impact on the propagation of water waves (Rey et al. 2014). This gave rise to new derivations of equations aimed to describe this interaction. First, an extended mild slope equation was obtained (Touboul et al. 2016). Then, the now classical coupled mode theory was introduced in the system to obtain a set of coupled equations, which could be compared to the system derived by Belibassakis et al (2011) but considering currents which may present constant shear with depth (Belibassakis et al. 2017, Belibassakis et al., 2019). In these works, the currents were assumed to vary linearly with depth, presenting a constant shear. However, this approach was recently extended to more general configurations (Belibassakis & Touboul, 2019; Touboul & Belibassakis, 2019).</p><p>In this work, we extend this model to three dimensional configurations. It is emphasized that the model is able to describe rotational waves, as expected, for example, when water waves propagate with a non-zero angle with respect to the current direction (see e.g. Ellingsen, 2016).</p><p>[1] Rey, V., Charland, J., Touboul, J., Wave – current interaction in the presence of a 3d bathymetry: deep water wave focusing in opposite current conditions. Phys. Fluids 26, 096601, 2014.</p><p>[2] Touboul J., Charland J., Rey V., Belibassakis K., Extended Mild-Slope equation for surface waves interacting with a vertically sheared current, Coastal Engineering, 116, 77–88, 2016.</p><p>[3] Belibassakis, K.A., Gerostathis, Th., Athanassoulis, G.A. A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography, Applied Ocean Res. 33, 384– 397, 2011.</p><p>[4] Belibassakis K.A., Simon B., Touboul J., Rey V., A coupled-mode model for water wave scattering by vertically sheared currents in variable bathymetry regions, Wave Motion, vol.74, 73-92, 2017.</p><p>[5] Belibassakis K., Touboul J., Laffitte E., Rey  V., A mild-slope system for Bragg scattering of water waves by sinusoidal bathymetry in the presence of vertically sheared currents,  J. Mar. Sci. Eng., Vol.7(1), 9, 2019.</p><p>[6] Belibassakis K.A., Touboul J. A nonlinear coupled-mode model for waves propagating in</p><p>vertically sheared currents in variable bathymetry-collinear waves and currents, Fluids, 4(2),</p><p>61, 2019.</p><p>[7] J. Touboul & K. Belibassakis, A novel method for water waves propagating in the presence of vortical mean flows over variable bathymetry, J. Ocean Eng. and Mar. Energy, https://doi.org/10.1007/s40722-019-00151-w, 2019.</p><p>[8] Ellingsen, S.A., Oblique waves on a vertically sheared current are rotational, Eur. J. Mech. B-Fluid 56, 156–160, 2016.</p>

scholarly journals WKB APPROXIMATION TO THE MODIFIED MILD-SLOPE EQUATION

2012 ◽  
Vol 1 (33) ◽  
pp. 3
Author(s):  
Seung-Nam Seo
Keyword(s):  
Shallow Water ◽  
Water Wave ◽  
Wave Scattering ◽  
Wkb Approximation ◽  
Linear Wave ◽  
Water Wave Scattering ◽  
Mild Slope Equation ◽  
Water Conditions ◽  
Rapidly Varying Topography ◽  
Present Equation

WKB approximation for water wave scattering by rapidly varying topography is obtained from a modified mild-slope equation of the general form by Porter (2003). The present WKB solution is reduced to the previous study where shallow water conditions are present. WKB models from the transformed mild-slope equation, without the described bottom curvature modification, show better performance than those by the original developed mild-slope equation. The underlying significance of the present equation is discussed in the context of linear wave scattering. The selected figures representing our results further characterize main feature of this study.

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