An exact analytic solution to the modified mild-slope equation for waves propagating over a trench with various shapes

Xie, Jian-Jian; Liu, Huan-Wen

doi:10.1016/j.oceaneng.2012.05.014

Cited by 28 publications

(2 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some authors studied the same problem by using the mild-slope or the modified mild-slope equation, providing analytical or semi-analytical solutions to reflection and transmission coefficients for typical trenches, valid in the linear, long-wave regime; see e.g. (Jung et al 2008;Michalsen et al 2008;Xie et al 2011;Xie and Liu 2012;H.-W. Liu et al 2013). Recent investigations, in the context of the fulllinear potential theory, have been presented by Mandal (2014, 2015), for normal and oblique incidence of waves over a single rectangular trench, respectively, and by Kar et al (2018), where a system of two 2D trenches of general shape has been considered.…”

Section: Introductionmentioning

confidence: 99%

A numerical study of the run-up and the force exerted on a vertical wall by a solitary wave propagating over two tandem trenches

Athanassoulis

Mavroeidis

Koutsogiannakis

et al. 2019

J. Ocean Eng. Mar. Energy

View full text Add to dashboard Cite

1. Introduction 2. An overview of the fully non-linear Hamiltonian Coupled-Mode Theory 2.1. Classical differential and variational formulation 2.2. The Hamiltonian Coupled-Mode Theory 3. A concise description of the numerical implementation 3.1. Numerical solution of the kinematical substrate problem 3.2. Numerical solution of the Hamiltonian evolution equations 4. Propagation of a solitary wave over some typical bathymetries. Validation and limitations of the present method 4.1. Propagation over a shelf 4.2. Propagation over a step 4.3. Comparison and assessment of the findings of Sec. 4.1 and 4.2 4.4. Propagation over a trench 5. Propagation of a solitary wave over two trenches and reflection at a vertical wall 5.1. Propagation over the trenches 5.2. Run-up on the wall 5.3. Maximum force exerted on the vertical wall Conclusions A numerical study of the run-up and the force exerted on a vertical wall by a solitary wave propagating over two tandem trenches and impinging on the wall AbstractThe propagation and transformation of water waves over varying bathymetries is a subject of fundamental interest to ocean, coastal and harbor engineers. The specific bathymetry considered in this paper consists of one or two, naturally formed or man-made, trenches. The problem we focus on is the transformation of an incoming solitary wave by the trench(es), and the impact of the resulting wave system on a vertical wall located after the trench(es). The maximum run-up and the maximum force exerted on the wall are calculated for various lengths and heights of the trench(es), and are compared with the corresponding quantities in the absence of them. The calculations have been performed by using the fully nonlinear water-wave equations, in the form of the Hamiltonian coupled-mode theory, recently developed in Papoutsellis et al (Eur. J. Mech. B/Fluids, Vol. 72, 2018, pp. 199-224). Comparisons of the calculated free-surface elevation with existing experimental results indicate that the effect of the vortical flow, inevitably developed within and near the trench(es) but not captured by any potential theory, is not important concerning the frontal wave flow regime. This suggests that the predictions of the run-up and the force on the wall by nonlinear potential theory are expected to be nearly realistic.The main conclusion of our investigation is that the presence of two tandem trenches in front of the wall may reduce the run-up from (about) 20% to 45% and the force from 15% to 38%., depending on the trench dimensions and the wave amplitude. The percentage reduction is greater for higher waves. The presence of only one trench leads to reductions 1.4 -1.7 times smaller.Keywords: nonlinear water waves; wave-trench interaction; solitary wave over varying bathymetry; run-up on vertical wall; force on vertical wall; submerged breakwater

show abstract

Section: Introductionmentioning

confidence: 99%

A numerical study of the run-up and the force exerted on a vertical wall by a solitary wave propagating over two tandem trenches

Athanassoulis

Mavroeidis

Koutsogiannakis

et al. 2019

J. Ocean Eng. Mar. Energy

View full text Add to dashboard Cite

show abstract

“…In this direction, Jung et al [15] obtained an approximated analytical solution for the mild-slope equation, applied to the analysis of wave reflection for an asymmetric trench. The aforementioned work was improved by Xie and Liu [16] by obtaining an exact analytical solution to the modified mild-slope equation. Bender and Dean [17] studied the reflection and transmission of normally incident waves by two-dimensional trenches and shoals.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Formulas for the Reflection/Transmission of Long Water Waves Propagating in a Tapered and Slender Harbor

Bautista

Méndez

Bautista

2015

Journal of Applied Mathematics

View full text Add to dashboard Cite

We obtain asymptotic formulas for the reflection/transmission coefficients of linear long water waves, propagating in a harbor composed of a tapered and slender region connected to uniform inlet and outlet regions. The region with variable character obeys a power-law. The governing equations are presented in dimensionless form. The reflection/transmission coefficients are obtained for the limit of the parameterκ2≪1, which corresponds to a wavelength shorter than the characteristic horizontal length of the harbor. The asymptotic formulas consider those cases when the geometry of the harbor can be variable in width and depth: linear or parabolic among other transitions or a combination of these geometries. For harbors with nonlinear transitions, the parabolic geometry is less reflective than the other cases. The reflection coefficient for linear transitions just presents an oscillatory behavior. We can infer that the deducted formulas provide as first approximation a practical reference to the analysis of wave reflection/transmission in harbors.

show abstract

Investigation of Solitary Wave Over a Trench Using a Non-hydrostatic Numerical Model

Zhang¹

2020

Springer Water

View full text Add to dashboard Cite

An exact analytic solution to the modified mild-slope equation for waves propagating over a trench with various shapes

Cited by 28 publications

References 21 publications

A numerical study of the run-up and the force exerted on a vertical wall by a solitary wave propagating over two tandem trenches

A numerical study of the run-up and the force exerted on a vertical wall by a solitary wave propagating over two tandem trenches

Asymptotic Formulas for the Reflection/Transmission of Long Water Waves Propagating in a Tapered and Slender Harbor

Investigation of Solitary Wave Over a Trench Using a Non-hydrostatic Numerical Model

Contact Info

Product

Resources

About