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Analytic solution to the modified mild-slope equation for reflection by a rectangular breakwater with scour trenches / Huan-Wen Liu in Journal of engineering mechanics, Vol. 139 N° 1 (Janvier 2013) 

Public ISBD [article]  in Journal of engineering mechanics > Vol. 139 N° 1 (Janvier 2013) . - pp.39–58. Titre : Analytic solution to the modified mild-slope equation for reflection by a rectangular breakwater with scour trenches Type de document : texte imprimé Auteurs : Huan-Wen Liu, Auteur ; Dan-Juan Fu, Auteur ; Xiao-Ling Sun, Auteur Année de publication : 2013 Article en page(s) : pp.39–58. Note générale : Mécanique appliquée Langues : Anglais (eng) Mots-clés : Breakwater  with  scour  trench,  Modified  mild-slope  equation  Analytic  solution  Reflection  coefficient  Periodicity  Periodic  oscillation Résumé : In this paper, an analytic solution to the modified mild-slope equation (MMSE) for wave reflection by a submerged rectangular breakwater with two scour trenches is explored. Because of the use of the MMSE with effects of the bottom curvature and the slope-squared terms, the solution is not only valid in the whole wave range from shallow water to deep water, but also valid for topographies not restricted to vary moderately. The present analytic solution includes an existing analytic long-wave solution as its special case, and the computing results show good agreement between two solutions, except for a slight difference when waves approach intermediate-wave range. It is found that this slight difference used to lead to an incorrect conclusion that the reflection coefficient for wave reflection by a rectangular breakwater or trench is a periodic function to the ratio of the breakwater length to the wavelength. This analysis shows that the reflection coefficient is a periodic oscillation function with a variable oscillation amplitude rather than a periodic function with a constant oscillation amplitude. It is also found that the discrepancy between the two solutions, respectively based on the MSE and the MMSE, mainly occurs for intermediate waves. Based on the present MMSE-based solution, the influence of trench dimensions on the reflection effect is investigated. It is shown that in the whole wave range, the phenomenon of zero reflection occurs more frequently for symmetrical bathymetry. ISSN : 0733-9399 En ligne : http://ascelibrary.org/doi/abs/10.1061/%28ASCE%29EM.1943-7889.0000481 [article] Analytic solution to the modified mild-slope equation for reflection by a rectangular breakwater with scour trenches [texte imprimé] / Huan-Wen Liu, Auteur ; Dan-Juan Fu, Auteur ; Xiao-Ling Sun, Auteur . - 2013 . - pp.39–58. Mécanique appliquée Langues : Anglais (eng) in Journal of engineering mechanics > Vol. 139 N° 1 (Janvier 2013) . - pp.39–58. Mots-clés : Breakwater  with  scour  trench,  Modified  mild-slope  equation  Analytic  solution  Reflection  coefficient  Periodicity  Periodic  oscillation Résumé : In this paper, an analytic solution to the modified mild-slope equation (MMSE) for wave reflection by a submerged rectangular breakwater with two scour trenches is explored. Because of the use of the MMSE with effects of the bottom curvature and the slope-squared terms, the solution is not only valid in the whole wave range from shallow water to deep water, but also valid for topographies not restricted to vary moderately. The present analytic solution includes an existing analytic long-wave solution as its special case, and the computing results show good agreement between two solutions, except for a slight difference when waves approach intermediate-wave range. It is found that this slight difference used to lead to an incorrect conclusion that the reflection coefficient for wave reflection by a rectangular breakwater or trench is a periodic function to the ratio of the breakwater length to the wavelength. This analysis shows that the reflection coefficient is a periodic oscillation function with a variable oscillation amplitude rather than a periodic function with a constant oscillation amplitude. It is also found that the discrepancy between the two solutions, respectively based on the MSE and the MMSE, mainly occurs for intermediate waves. Based on the present MMSE-based solution, the influence of trench dimensions on the reflection effect is investigated. It is shown that in the whole wave range, the phenomenon of zero reflection occurs more frequently for symmetrical bathymetry. ISSN : 0733-9399 En ligne : http://ascelibrary.org/doi/abs/10.1061/%28ASCE%29EM.1943-7889.0000481

Analytical solution for long - wave reflection by a general breakwater or trench with curvilinear slopes / Huan-Wen Liu in Journal of engineering mechanics, Vol. 139 N° 2 (Février 2013) 

Public ISBD [article]  in Journal of engineering mechanics > Vol. 139 N° 2 (Février 2013) . - pp.229–245. Titre : Analytical solution for long - wave reflection by a general breakwater or trench with curvilinear slopes Type de document : texte imprimé Auteurs : Huan-Wen Liu, Auteur ; Jiong-Xing Luo, Auteur ; Pengzhi Lin, Auteur Année de publication : 2013 Article en page(s) : pp.229–245. Note générale : Applied mechanics Langues : Anglais (eng) Mots-clés : Long-wave  equation  General  breakwater  or  trench  Curvilinear  slope  Reflection  coefficient  Closed-form  solution  Series  solution Résumé : In the first part of this paper, an exact analytical solution in closed form for linear long-wave reflection by a submerged idealized breakwater or trench with various curvilinear slopes is given. The solution obtained finds almost all previous long-wave analytical solutions for wave reflection by idealized bathymetries to be its special cases, including the wave reflection by an infinite step, a continental shelf with a parabolic slope, a continental shelf with a linear slope, a rectangular obstacle, an obstacle of general trapezoidal shape with linear slopes, and a trench of general trapezoidal shape with linear slopes. In the second part, an exact analytical solution in the form of a Taylor series for linear long-wave reflection by a submerged quasi-idealized breakwater or trench is also constructed. It is shown by convergence analysis that the series solution converges in the entire physical domain. Based on the present analytical solutions, the reflection coefficients for long waves reflected by various breakwaters are calculated and the influence of the breakwater dimensions in the reflection effect is investigated. It is always found that the total reflection defined by the area under the reflection coefficient curve increases when the front and back slopes become steep. It is also found that the phenomenon of zero reflection for a symmetrical rectangular breakwater still remains for a general breakwater with curvilinear slopes as long as the bathymetry is symmetrical about the breakwater. ISSN : 0733-9399 En ligne : http://ascelibrary.org/doi/abs/10.1061/%28ASCE%29EM.1943-7889.0000483 [article] Analytical solution for long - wave reflection by a general breakwater or trench with curvilinear slopes [texte imprimé] / Huan-Wen Liu, Auteur ; Jiong-Xing Luo, Auteur ; Pengzhi Lin, Auteur . - 2013 . - pp.229–245. Applied mechanics Langues : Anglais (eng) in Journal of engineering mechanics > Vol. 139 N° 2 (Février 2013) . - pp.229–245. Mots-clés : Long-wave  equation  General  breakwater  or  trench  Curvilinear  slope  Reflection  coefficient  Closed-form  solution  Series  solution Résumé : In the first part of this paper, an exact analytical solution in closed form for linear long-wave reflection by a submerged idealized breakwater or trench with various curvilinear slopes is given. The solution obtained finds almost all previous long-wave analytical solutions for wave reflection by idealized bathymetries to be its special cases, including the wave reflection by an infinite step, a continental shelf with a parabolic slope, a continental shelf with a linear slope, a rectangular obstacle, an obstacle of general trapezoidal shape with linear slopes, and a trench of general trapezoidal shape with linear slopes. In the second part, an exact analytical solution in the form of a Taylor series for linear long-wave reflection by a submerged quasi-idealized breakwater or trench is also constructed. It is shown by convergence analysis that the series solution converges in the entire physical domain. Based on the present analytical solutions, the reflection coefficients for long waves reflected by various breakwaters are calculated and the influence of the breakwater dimensions in the reflection effect is investigated. It is always found that the total reflection defined by the area under the reflection coefficient curve increases when the front and back slopes become steep. It is also found that the phenomenon of zero reflection for a symmetrical rectangular breakwater still remains for a general breakwater with curvilinear slopes as long as the bathymetry is symmetrical about the breakwater. ISSN : 0733-9399 En ligne : http://ascelibrary.org/doi/abs/10.1061/%28ASCE%29EM.1943-7889.0000483

Analytical solution for long - wave reflection by a rectangular obstacle with two scour trenches / Jian-Jian Xie in Journal of engineering mechanics, Vol. 137 N° 12 (Decembre 2011) 

Public ISBD [article]  in Journal of engineering mechanics > Vol. 137 N° 12 (Decembre 2011) . - pp.919-930 Titre : Analytical solution for long - wave reflection by a rectangular obstacle with two scour trenches Type de document : texte imprimé Auteurs : Jian-Jian Xie, Auteur ; Huan-Wen Liu, Auteur ; Pengzhi Lin, Auteur Année de publication : 2012 Article en page(s) : pp.919-930 Note générale : Mécanique appliquée Langues : Anglais (eng) Mots-clés : Scour  Long  waves  Wave  reflection  Coefficients Résumé : In this paper, linear long-wave reflection by a rectangular obstacle with two scour trenches of power function profile is explored. A closed-form analytical solution in terms of the first and second kinds of Bessel functions is obtained, which finds two classic analytical solutions as its special cases, i.e., the wave reflection from a rectangular obstacle and from an infinite step. The phenomenon of zero reflection coefficient for a single rectangular obstacle with the same depths in front of and behind the obstacle still remains for a rectangular obstacle with two scour trenches as long as the bathymetry is symmetrical about the obstacle. The periodicity of the reflection coefficient as a function of the relative length of the middle rectangular obstacle disappears if two scour trenches are attached to the middle rectangular obstacle. Finally, the wave reflection by a rectangular obstacle with two scour trenches generally increases when the trenches become wide and deep. The wave reflection by a degenerated single slope increases when the slope becomes deep. DEWEY : 620.1 ISSN : 0733-9399 En ligne : http://ascelibrary.org/emo/resource/1/jenmdt/v137/i12/p919_s1?isAuthorized=no [article] Analytical solution for long - wave reflection by a rectangular obstacle with two scour trenches [texte imprimé] / Jian-Jian Xie, Auteur ; Huan-Wen Liu, Auteur ; Pengzhi Lin, Auteur . - 2012 . - pp.919-930. Mécanique appliquée Langues : Anglais (eng) in Journal of engineering mechanics > Vol. 137 N° 12 (Decembre 2011) . - pp.919-930 Mots-clés : Scour  Long  waves  Wave  reflection  Coefficients Résumé : In this paper, linear long-wave reflection by a rectangular obstacle with two scour trenches of power function profile is explored. A closed-form analytical solution in terms of the first and second kinds of Bessel functions is obtained, which finds two classic analytical solutions as its special cases, i.e., the wave reflection from a rectangular obstacle and from an infinite step. The phenomenon of zero reflection coefficient for a single rectangular obstacle with the same depths in front of and behind the obstacle still remains for a rectangular obstacle with two scour trenches as long as the bathymetry is symmetrical about the obstacle. The periodicity of the reflection coefficient as a function of the relative length of the middle rectangular obstacle disappears if two scour trenches are attached to the middle rectangular obstacle. Finally, the wave reflection by a rectangular obstacle with two scour trenches generally increases when the trenches become wide and deep. The wave reflection by a degenerated single slope increases when the slope becomes deep. DEWEY : 620.1 ISSN : 0733-9399 En ligne : http://ascelibrary.org/emo/resource/1/jenmdt/v137/i12/p919_s1?isAuthorized=no

Analytical Study of Linear Long-Wave Reflection by a Two-Dimensional Obstacle of General Trapezoidal Shape / Pengzhi Lin in Journal of engineering mechanics, vol.131, N° 8 (Aout 2005)

Public ISBD [article]  in Journal of engineering mechanics > vol.131, N° 8 (Aout 2005) . - 822-830 p. Titre : Analytical Study of Linear Long-Wave Reflection by a Two-Dimensional Obstacle of General Trapezoidal Shape Titre original : L'Etude Analytique de Linéaire Long-Ondulent la Réflexion par un Obstacle Deux-Dimensionnel de Forme Trapézoïdale Générale Type de document : texte imprimé Auteurs : Pengzhi Lin, Auteur ; Huan-Wen Liu, Auteur Article en page(s) : 822-830 p. Note générale : Génie Mécanique, Génie civil Langues : Anglais (eng) Mots-clés : Wave  reflection  Long  waves  Breakwaters  Analytical  techniques  Reflexion  d'onde  Longues  vagues  Brise-lames  Techniques  analytiques Index. décimale : 621.34/624 Résumé : In This Paper an analytical solution for linear long-wave reflection by an obstacle of general trapezoidal shape is explored. A Closed-form expression in terms of fist and second kinds of bessel functions is obtained for the wave reflection coefficient, which depends on the relative lengths of the two slopes and top of the obstacle as well as the depth ratios in front of and behind the obstacle versus that above the obstacle.That Analytical solution obtained in this study finds a few well-known analytical solutions to be its special cases, which include the wave reflection from a rectangular obstacle , an infinite step, and an infinite step behind a linear slope. The Present analytical solution, however, covers a much wider range of problems. It Is found that the periodicity of the wave reflection coefficient as the function of the relative length of the obstacle remains when two slopes are present but with a reduced magnitude.The Phenomenon of Zero wave reflection from the structure is special to a rectangular obstacle only, which disapears with the addition of a slope in front or at the rear.The New solution may be very useful in some engieering applications, for example the design of a submerged breakwater of trapezoidal shape. En cet article que une solution analytique pour linéaire long-ondulent la réflexion par un obstacle de forme trapézoïdale générale est exploré. Une expression de forme close en termes de poing et les deuxièmes genres de fonctions de bessel est obtenue pour le coefficient de réflexion de vague, devant lequel dépend des longueurs relatives des deux pentes et le dessus de l'obstacle aussi bien que les rapports de profondeur et derrière l'obstacle contre cela au-dessus de la solution analytique d'obstacle.That obtenue en trouvailles de cette étude quelques solutions analytiques bien connues d'être ses cas spéciaux, qui incluent la réflexion de vague d'un obstacle rectangulaire, étape infinie, et une étape infinie derrière une pente linéaire. La solution analytique actuelle, cependant, couvre un éventail beaucoup de problèmes. On le constate que la périodicité du coefficient de réflexion de vague comme fonction de la longueur relative de l'obstacle demeure quand deux pentes sont présentes mais avec un phénomène réduit de magnitude.The de la réflexion zéro de vague de la structure est spécial à un obstacle rectangulaire seulement, qui les disapears avec l'addition d'une pente dans l'avant ou à la nouvelle solution de rear.The peuvent être très utiles dans quelques applications engieering, par exemple la conception d'un brise-lames submergé de forme trapézoïdale. [article] Analytical Study of Linear Long-Wave Reflection by a Two-Dimensional Obstacle of General Trapezoidal Shape = L'Etude Analytique de Linéaire Long-Ondulent la Réflexion par un Obstacle Deux-Dimensionnel de Forme Trapézoïdale Générale [texte imprimé] / Pengzhi Lin, Auteur ; Huan-Wen Liu, Auteur . - 822-830 p. Génie Mécanique, Génie civil Langues : Anglais (eng) in Journal of engineering mechanics > vol.131, N° 8 (Aout 2005) . - 822-830 p. Mots-clés : Wave  reflection  Long  waves  Breakwaters  Analytical  techniques  Reflexion  d'onde  Longues  vagues  Brise-lames  Techniques  analytiques Index. décimale : 621.34/624 Résumé : In This Paper an analytical solution for linear long-wave reflection by an obstacle of general trapezoidal shape is explored. A Closed-form expression in terms of fist and second kinds of bessel functions is obtained for the wave reflection coefficient, which depends on the relative lengths of the two slopes and top of the obstacle as well as the depth ratios in front of and behind the obstacle versus that above the obstacle.That Analytical solution obtained in this study finds a few well-known analytical solutions to be its special cases, which include the wave reflection from a rectangular obstacle , an infinite step, and an infinite step behind a linear slope. The Present analytical solution, however, covers a much wider range of problems. It Is found that the periodicity of the wave reflection coefficient as the function of the relative length of the obstacle remains when two slopes are present but with a reduced magnitude.The Phenomenon of Zero wave reflection from the structure is special to a rectangular obstacle only, which disapears with the addition of a slope in front or at the rear.The New solution may be very useful in some engieering applications, for example the design of a submerged breakwater of trapezoidal shape. En cet article que une solution analytique pour linéaire long-ondulent la réflexion par un obstacle de forme trapézoïdale générale est exploré. Une expression de forme close en termes de poing et les deuxièmes genres de fonctions de bessel est obtenue pour le coefficient de réflexion de vague, devant lequel dépend des longueurs relatives des deux pentes et le dessus de l'obstacle aussi bien que les rapports de profondeur et derrière l'obstacle contre cela au-dessus de la solution analytique d'obstacle.That obtenue en trouvailles de cette étude quelques solutions analytiques bien connues d'être ses cas spéciaux, qui incluent la réflexion de vague d'un obstacle rectangulaire, étape infinie, et une étape infinie derrière une pente linéaire. La solution analytique actuelle, cependant, couvre un éventail beaucoup de problèmes. On le constate que la périodicité du coefficient de réflexion de vague comme fonction de la longueur relative de l'obstacle demeure quand deux pentes sont présentes mais avec un phénomène réduit de magnitude.The de la réflexion zéro de vague de la structure est spécial à un obstacle rectangulaire seulement, qui les disapears avec l'addition d'une pente dans l'avant ou à la nouvelle solution de rear.The peuvent être très utiles dans quelques applications engieering, par exemple la conception d'un brise-lames submergé de forme trapézoïdale.