Analytic solution of long wave propagation over a submerged hump

Niu, Xiaojing; Yu, Xiping

doi:10.1016/j.coastaleng.2010.09.001

Cited by 21 publications

(12 citation statements)

References 7 publications

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“…The Green-Naghdi equations follow when taking an appropriate average of vertical variables [20][21][22]. Another study of long wave propagation over a submerged 2-dimensional bump was recently presented in [23], albeit according to linear longwave theory.…”

Section: Introductionmentioning

confidence: 99%

Shallow-water soliton dynamics beyond the Korteweg–de Vries equation

2014

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An alternative way for the derivation of the new KdV-type equation is presented. The equation contains terms depending on the bottom topography (there are six new terms in all, three of which are caused by the unevenness of the bottom). It is obtained in the second order perturbative approach in the weakly nonlinear, dispersive and long wavelength limit. Only treating all these terms in the second order perturbation theory made the derivation of this KdV-type equation possible. The motion of a wave, which starts as a KdV soliton, is studied according to the new equation in several cases by numerical simulations. The quantitative changes of a soliton's velocity and amplitude appear to be directly related to bottom variations. Changes of the soliton's velocity appear to be almost linearly anticorrelated with changes of water depth whereas correlation of variation of soliton's amplitude with changes of water depth looks less linear. When the bottom is flat, the new terms narrow down the family of exact solutions, but at least one single soliton survives. This is also checked by numerics.

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Section: Introductionmentioning

confidence: 99%

Shallow-water soliton dynamics beyond the Korteweg–de Vries equation

2014

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“…As the present solution can be used to approximate the water wave propagation over a pit or hump on the uniform water depth, it is compared with the solution for a submerged hump reported by Niu and Yu (2011a) and the solution for an axisymmetric pit by Jung and Suh (2007). The influence of the topographic parameters of the seamount on wave scattering is investigated in detail using the analytical solution.…”

Section: Discussionmentioning

confidence: 99%

“…Niu and Yu (2011a) derived an analytic solution for long wave refraction by a submerged circular hump, and presented results for h 1 = 3.2 m, h 2 = 4.8 m, k 2 h 2 = 0.3 and r 2 = π/k 2 . The current solution with h 0 = 1.01h 1 , r 1 = 0.5 m and m = 4 is compared with results of Niu and Yu (2011a) for α = 1. The overall good agreement is obtained between the two sets of solutions as shown in Figure 5.…”

Section: Comparison With Existing Solutionmentioning

confidence: 99%

“…Zhu and Harun (2009) further derived an analytic solution of long-wave over a quasi-ideal parabolic shoal defined by the water-depth function h(r) = ar 2 +h 0 (where h 0 is the minimum water depth of the hump). Niu and Yu (2011a) and Liu and Xie (2011) generalized the shoal with power function profiles h(r) = ar m + h 0 (m is any positive integer) and presented the analytic solution to explore the impact of the shape, size, height of shoal on the focus position and the maximum wave amplitude.…”

Section: Introductionmentioning

confidence: 99%

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Analytic study on long wave transformation over a seamount with a pit

Wang

Zheng

et al. 2018

Ocean Engineering

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In this paper, an analytic solution is derived for linear long waves scattering over a submarine seamount landform with a pit. The seamount is axisymmetric with a pit on the top. The water depth is defined by a trinomial function in the radial direction. The governing linear shallow water equation for long waves is expressed in the polar coordination, which is solved through separation of variables. As the topography is axisymmetric, solutions can be written as Fourier-cosine series. Waves over the seamount are expressed using Frobenius series expansion, while the water surface elevation in the outer region is expressed as Fourier-Bessel series, and the final solution is obtained by matching them at the conjunction. The solution can be degenerated into the previous analytic solutions for waves propagation over an axisymmetric pit or a submerged hump by adjusting the topography parameters.

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“…The length of the square computational domain is 6r 0 and the center of the shoal is located at ð0; 0Þ. The analytical solution is referred to Niu and Yu (2011) and plotted in Fig. 11.…”

Section: Shallow Water Waves Propagating Over a Submerged Shoalmentioning

confidence: 99%