A high-order cnoidal wave theory

1979 ◽  
Vol 94 (1) ◽  
pp. 129-161 ◽  
Author(s):  
J. D. Fenton
Keyword(s):  
Shallow Water ◽  
Wave Height ◽  
Water Depth ◽  
Wave Theory ◽  
High Order ◽  
Stokes Wave ◽  
Cnoidal Wave ◽  
Expansion Parameter ◽  
Wave Solutions ◽  
Fifth Order

A method is outlined by which high-order solutions are obtained for steadily progressing shallow water waves. It is shown that a suitable expansion parameter for these cnoidal wave solutions is the dimensionless wave height divided by the parameter m of the cn functions: this explicitly shows the limitation of the theory to waves in relatively shallow water. The corresponding deep water limitation for Stokes waves is analysed and a modified expansion parameter suggested.Cnoidal wave solutions to fifth order are given so that a steady wave problem with known water depth, wave height and wave period or length may be solved to give expressions for the wave profile and fluid velocities, as well as integral quantities such as wave power and radiation stress. These series solutions seem to exhibit asymptotic behaviour such that there is no gain in including terms beyond fifth order. Results from the present theory are compared with exact numerical results and with experiment. It is concluded that the fifth-order cnoidal theory should be used in preference to fifth-order Stokes wave theory for wavelengths greater than eight times the water depth, when it gives quite accurate results.

Conditional Short-Crested Waves in Shallow Water and With Superimposed Current

2002 ◽  
Author(s):  
Jo̸rgen Juncher Jensen
Keyword(s):  
Shallow Water ◽  
Water Depth ◽  
Wave Spectrum ◽  
Wave Theory ◽  
Ocean Waves ◽  
Offshore Structures ◽  
Stokes Wave ◽  
Wave Crest ◽  
Non Linear ◽  
Drilling Rigs

For bottom-supported offshore structures like oil drilling rigs and oil production platforms, a deterministic design wave approach is often applied using a regular non-linear Stokes’ wave. Thereby, the procedure accounts for non-linear effects in the wave loading but the randomness of the ocean waves is poorly represented, as the shape of the wave spectrum does not enter the wave kinematics. To overcome this problem and still keep the simplicity of a deterministic approach, Tromans, Anaturk and Hagemeijer (1991) suggested the use of a deterministic wave, defined as the expected linear Airy wave, given the value of the wave crest at a specific point in time or space. In the present paper a derivation of the expected linear short-crested wave riding on a uniform current is given. The analysis is based on the conventional shallow water Airy wave theory and the direction of the main wind direction can make any direction with the current. A consistent derivation of the wave spectrum taking into account current and finite water depth is used. The numerical results show a significant effect of the water depth, the directional spreading and the current on the conditional mean wave profile. Extensions to higher order waves are finally discussed.

Cnoidal Wave-Induced Residual Liquefaction in Loosely Deposited Seabed under Coastal Shallow Water

2021 ◽  
Vol 11 (24) ◽  
pp. 11631
Author(s):  
Xiuwei Chai ◽  
Jingyuan Liu ◽  
Yu Zhou
Keyword(s):  
Shallow Water ◽  
Pore Pressure ◽  
Water Environment ◽  
Lateral Spreading ◽  
Stokes Wave ◽  
Cnoidal Wave ◽  
Liquefaction Resistance ◽  
Liquefaction Process ◽  
Wave Induced ◽  
Seabed Soil

This study is aimed at numerically investigating the cnoidal wave-induced dynamics characteristics and the liquefaction process in a loosely deposited seabed floor in a shallow water environment. To achieve this goal, the integrated model FSSI-CAS 2D is taken as the computational platform, and the advanced soil model Pastor–Zienkiewicz Mark III is utilized to describe the complicated mechanical behavior of loose seabed soil. The computational results show that a significant lateral spreading and vertical subsidence could be observed in the loosely deposited seabed floor due to the gradual loss of soil skeleton stiffness caused by the accumulation of pore pressure. The accumulation of pore pressure in the loose seabed is not infinite but limited by the liquefaction resistance line. The seabed soil at some locations could be reached to the full liquefaction state, becoming a type of heavy fluid with great viscosity. Residual liquefaction is a progressive process that is initiated at the upper part of the seabed floor and then enlarges downward. For waves with great height in shallow water, the depth of the liquefaction zone will be greatly overestimated if the Stokes wave theory is used. This study can enhance the understanding of the characteristics of the liquefaction process in a loosely deposited seabed under coastal shallow water and provide a reference for engineering activities.

Effects of Cnoidal Wave-Induced Seepage on Vertical Breakwater Resting on Permeable Elastic Seabed

2014 ◽  
Vol 716-717 ◽  
pp. 284-288
Author(s):  
Jian Kang Yang ◽  
Hua Huang ◽  
Lin Guo ◽  
Jing Rong Lin ◽  
Qing Yong Zhu ◽  
...  
Keyword(s):  
Shallow Water ◽  
Wave Theory ◽  
Cnoidal Wave ◽  
Load Prediction ◽  
Wave Load ◽  
Seepage Pressure ◽  
Theoretical Investigations ◽  
Nonlinearity Effect ◽  
Wave Nonlinearity ◽  
Wave Induced

Theoretical investigations on cnoidal waves interacting with breakwater resting on permeable elastic seabed are presented in this paper. Based on the shallow water reflected wave theory and Biot consolidation theory on wave-induced seepage pressure, the analytical solutions to first order cnoidal wave reflection and wave-induced seepage pressure are obtained by the eigenfunction expansion approach. Numerical results are presented to show the effects of depth of water, breakwater geometry on cnoidal wave-induced seepage uplift force and overturning moment. Compared with Airy wave theory, in certain shallow water conditions, the shallow water wave theory can more effectively illustrate wave nonlinearity effect in wave load prediction.

Exact Solitary-wave Solutions for the General Fifth-order Shallow Water-wave Models

1999 ◽  
Vol 54 (3-4) ◽  
pp. 272-274
Author(s):  
Woo-Pyo Hong ◽  
Young-Dae Jung
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Symbolic Computation ◽  
Water Wave ◽  
Model Parameters ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Special Cases ◽  
Wave Models ◽  
Fifth Order

We perform a computerized symbolic computation to find some general solitonic solutions for the general fifth-order shal-low water-wave models. Applying the tanh-typed method, we have found certain new exact solitary wave solutions. The pre-viously published solutions turn out to be special cases with restricted model parameters.

scholarly journals A Study on the Minimization of Mooring Load in Fish-Cage Mooring Systems with a Damping Buoy

2020 ◽  
Vol 8 (10) ◽  
pp. 814
Author(s):  
Gun-Ho Lee ◽  
Bong-Jin Cha ◽  
Hyun-young Kim
Keyword(s):  
Numerical Simulations ◽  
Wave Theory ◽  
Line Length ◽  
Stokes Wave ◽  
Mooring Line ◽  
Spring Model ◽  
Mass Spring Model ◽  
Wave Conditions ◽  
Mass Spring ◽  
Fifth Order

This study established the conditions in which mooring load is minimized in a fish cage that includes a damping buoy in specific wave conditions. To derive these conditions, numerical simulations of various mooring contexts were conducted on a fish cage (1/15 scale) using a simplified mass-spring model and fifth-order Stokes wave theory. The simulation conditions were as follows: (1) bridle-line length of 0.8–3.2 m; (2) buoyancy of 2.894–20.513 N for the damping buoy; and (3) mooring-rope thickness of 0.002–0.004 m. The wave conditions were 0.333 m in height and 1.291–2.324 s of arrival period. Consequently, the mooring tensions tended to decrease with decreasing mooring line thickness and increasing bridle-line length and buoyancy of the buoy. Accordingly, it was assumed to be advantageous to minimize the mooring tension by designing a thin mooring line and long bridle line and for the buoyancy of the buoy to be as large as possible. This approach shows a valuable technique because it can contribute to the improvement of the mooring stability of the fish cage by establishing a method that can be used to minimize the load on the mooring line.

Contributions to the theory of stokes waves

1955 ◽  
Vol 51 (4) ◽  
pp. 713-736 ◽  
Author(s):  
S. C. De
Keyword(s):  
Wave Height ◽  
Flow Rate ◽  
Volume Flow Rate ◽  
Wave Theory ◽  
Finite Depth ◽  
Volume Flow ◽  
Steady Flows ◽  
Total Head ◽  
Stokes Waves ◽  
Fifth Order

ABSTRACTThe well-known Stokes theory (9, 10) of waves of permanent form in water of finite depth has been extended to the fifth order of approximation. The solutions have been first obtained in the form of equations for the space coordinates x and y as functions of the velocity potential Φ and stream function ψ. Expressions for the complex potential W in terms of the complex variable z ( = x + iy), the form of the wave profile, and the square of the wave velocity have been obtained to the fifth order.Expressions for the three physical quantities Q, R and S, where Q is the volume flow rate per unit span, R is the energy per unit mass (i.e. g times the total head, measuring heights from the bottom and pressures from atmospheric) and S is the momentum flow rate per unit spaa, corrected for pressure forces and divided by density, have been obtained to the fifth order. The values for the dimensionless quantities r = R/Rc and s = S/Sc, where Rc and Sc refer to the values of R and S for a critical stream of volume flow Q, are tabulated for certain values of the ratios mean depth to wavelength and amplitude to wavelength. The values of r and s thus obtained have been used to calculate the ratios of mean depth to wavelength and of wave height to wavelength according to the cnoidal wave theory as recently presented by Benjamin and Light-hill(1), and the results are found to be in satisfactory agreement with that from Stokes's theory for waves longer than six times the depth.The (r, s) diagram introduced in the recent work of Benjamin and Lighthill(1) has been further considered, and the unshaded part of the diagram referred to in that paper has been mapped with a network of curves for constant values of the ratios of mean depth to wavelength and of wave height to wavelength (Fig. 2). The third barrier to the existence of steady flows, corresponding to ‘waves of greatest height’ referred to in that paper, has also been indicated in Fig. 2.

High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations

2017 ◽  
Vol 22 (4) ◽  
pp. 1049-1068 ◽  
Author(s):  
Zhen Gao ◽  
Guanghui Hu
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
High Order ◽  
High Order Accuracy ◽  
Local Characteristic ◽  
Order Accuracy ◽  
Splitting Technique ◽  
Strong Discontinuities ◽  
Comput Phys ◽  
Fifth Order

AbstractIn this study, a numerical framework of the high order well-balanced weighted compact nonlinear (WCN) schemes is proposed for the shallow water equations based on the work in [S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321]. We employ a special splitting technique for the source term proposed in [Y. Xing, C.-W Shu, J. Comput. Phys. 208 (2005) 206-227] to maintain the exact C-property, which can be proved theoretically. In the meantime, the genuine high order accuracy of the numerical scheme can be observed successfully, and small perturbation of the stationary state can be resolved and evolved well. In order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact scheme are used to construct different WCN schemes. In addition, the local characteristic projections are considered to further restrain the potential numerical oscillations. A variety of representative one- and two-dimensional examples are tested to demonstrate the good performance of the proposed schemes.

scholarly journals EFFECT OF BEACH SLOPE AND SHOALING ON WAVE ASYMMETRY

1968 ◽  
Vol 1 (11) ◽  
pp. 10 ◽  
Author(s):  
M.D. Adeyemo
Keyword(s):  
Shallow Water ◽  
Quantitative Study ◽  
Experimental Work ◽  
Theoretical Study ◽  
Wave Theory ◽  
Cnoidal Wave ◽  
Quantitative Correlation ◽  
Wave Slope ◽  
Oscillatory Waves ◽  
Good Agreement

This paper is concerned with the quantitative study of the geometrical asymmetry associated with shallow water oscillatory waves in the breaker zone. Three descriptions of wave asymmetry are defined and examined : (l) Wave vertical asymmetry i2^ Wave slope asymmetry and (3) Wave horizontal asymmetry The effects of shoaling, produced by beaches of different slope, on the wave asymmetry are examined. Six beach slopes in the range 1:4 to 1:18 were employed, and a quantitative correlation was found to exist between the wave slope asymmetry, wave horizontal asymmetry and the wave vertical asymmetry. An expression is given for the wave horizontal asymmetry based on the expression for the wave vertical asymmetry from the cnoidal wave theory. The theoretical study of wave slope asymmetry made by Biesel (1) and the results of the experimental work on the wave slope asymmetry in the present work are compared and gave a good agreement*

scholarly journals THE MEASURED PROPERTIES OF IRREGULAR WAVE BREAKING AND WAVE HEIGHT CHANGE AFTER BREAKING ON THE SLOPE

1988 ◽  
Vol 1 (21) ◽  
pp. 29 ◽  
Author(s):  
Akira Seyama ◽  
Akira Kimura
Keyword(s):  
Wave Height ◽  
Water Depth ◽  
Wave Theory ◽  
Irregular Waves ◽  
Linear Wave ◽  
Periodic Waves ◽  
Zero Crossing ◽  
Irregular Wave ◽  
Height Change ◽  
Cross Waves

Wave height change of the zero-down-cross waves on uniform slopes were examined experimentally. The properties of shoaling, breaking and decay after breaking for a total of about 4,000 irregular waves of the Pierson-Moskowitz type on 4 different slopes (1/10, 1/20, 1/30 and 1/50) were investigated. The shoaling property of the zero-down-cross waves can be approximated by the linear wave theory. However, the properties of breaking and decay after breaking differ considerably from those for periodic waves. The wave height water depth ratio (H/d) at the breaking point for the zero-down-cross waves is about 30% smaller than that for periodic waves on average despite the slopes. Wave height decay after breaking also differs from that for periodic waves and can be classified into three regions, i.e. shoaling, plunging and bore regions. Experimental equations for the breaking condition and wave height change after breaking are proposed in the study. A new definition of water depth for the zero-crossing wave analysis which can reduce the fluctuation in the plotted data is also proposed.

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