# Formation of Three-Dimensional Surface Waves on Deep-Water Using Elliptic Solutions of Nonlinear Schrödinger Equation

@article{Sajjadi2015FormationOT, title={Formation of Three-Dimensional Surface Waves on Deep-Water Using Elliptic Solutions of Nonlinear Schr{\"o}dinger Equation}, author={Shahrdad G. Sajjadi and Stefan C. Mancas and Fr{\'e}d{\'e}rique Drullion}, journal={Advances and Applications in Fluid Mechanics}, year={2015}, volume={18}, pages={81-112} }

A review of three-dimensional waves on deep-water is presented. Three forms of three dimensionality, namely oblique, forced and spontaneous type, are identified. An alternative formulation for these three-dimensional waves is given through cubic nonlinear Schr\"odinger equation. The periodic solutions of the cubic nonlinear Schr\"odinger equation are found using Weierstrass elliptic $\wp$ functions. It is shown that the classification of solutions depends on the boundary conditions, wavenumber…

## Figures from this paper

## 2 Citations

### EXACT ANALYTICAL SOLUTION OF VISCOUS KORTEWEG-DE VRIES EQUATION FOR WATER WAVES

- Physics, Mathematics
- 2016

The evolution of a solitary wave with very weak nonlinearity which was originally investigated by Miles [4] is revisited. The solution for a one-dimensional gravity wave in a water of uniform depth…

### Traveling Wave Solutions to Kawahara and Related Equations

- Mathematics
- 2016

Traveling wave solutions to Kawahara equation (KE), transmission line (TL), and Korteweg–de Vries (KdV) equation are found by using an elliptic function method which is more general than the…

## References

SHOWING 1-10 OF 30 REFERENCES

### Three-dimensional deep-water waves. Part 1. Experimental measurement of skew and symmetric wave patterns

- PhysicsJournal of Fluid Mechanics
- 1982

The three-dimensional structures of symmetric and skew wave patterns in deep water observed in a tow tank and a wide basin are described. These symmetric waves are the result of three-dimensional…

### A new type of three-dimensional deep-water wave of permanent form

- PhysicsJournal of Fluid Mechanics
- 1980

A new class of three-dimensional, deep-water gravity waves of permanent form has been found using an equation valid for weakly nonlinear waves due to Zakharov (1968). These solutions appear as…

### Stability of periodic waves of finite amplitude on the surface of a deep fluid

- Physics
- 1968

We study the stability of steady nonlinear waves on the surface of an infinitely deep fluid [1, 2]. In section 1, the equations of hydrodynamics for an ideal fluid with a free surface are transformed…

### Notes on long-crested water waves

- MathematicsJournal of Fluid Mechanics
- 1983

Fully three-dimensional surface gravity waves in deep water are investigated in the limit in which the length of the wave crests become long. We describe an analytic solution to fourth order in wave…

### A semi-analytic solution for nonlinear standing waves in deep water

- MathematicsJournal of Fluid Mechanics
- 1981

A method of time-dependent conformal mapping is introduced to simplify the power-series solution procedure for time- and space-periodic standing waves in deep water. A solution has been found to 25th…

### On the generation of surface waves by shear flows

- Environmental Science, PhysicsJournal of Fluid Mechanics
- 1957

A mechanism for the generation of surface waves by a parallel shear flow U(y) is developed on the basis of the inviscid Orr-Sommerfeld equation. It is found that the rate at which energy is…

### The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics

- PhysicsProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1978

In this paper we embark on a calculation of all the normal-mode perturbations of nonlinear, irrotational gravity waves as a function of the wave steepness. The method is to use as coordinates the…

### Asymptotic multi-layer analysis of wind over unsteady monochromatic surface waves

- Environmental Science
- 2013

Asymptotic multi-layer (AML) analyses and computation of solutions for turbulent flows over steady and unsteady monochromatic surface waves are reviewed in the limits of low-turbulence stresses and…

### A note on some nonlinear water-wave experiments and the comparison of data with theory

- PhysicsJournal of Fluid Mechanics
- 1977

The problem of the instability of a uniform, nonlinear, deep-water wave train to infinitesimal long-wave perturbations, first studied by Benjamin ' Feir (1967) and Benjamin (1967), is re-examined. It…