Oblique runup of non-breaking solitary waves on an inclined plane

2011 ◽  
Vol 668 ◽  
pp. 582-606 ◽  
Author(s):  
GEIR K. PEDERSEN
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Evolution Equations ◽  
Boussinesq Equations ◽  
Normal Incidence ◽  
Finite Depth ◽  
Breaking Waves ◽  
Wave Pattern ◽  
Angle Of Incidence ◽  
Inclined Plane

When a wave of permanent form is obliquely incident on an inclined plane, the wave pattern becomes stationary in a frame of reference which moves along the shore. This enables a simplified mathematical description of the problem which is used herein as a basis for efficient and accurate numerical simulations. First, a nonlinear and weakly dispersive set of Boussinesq equations for the downstream evolution of such stationary patterns is derived. In the hydrostatic approximation, streamline-based Lagrangian versions of the evolution equations are developed for automatic tracing of the shoreline. Both equation sets are, in their present form, developed for non-breaking waves only. Finite difference models for both equation sets are designed. These methods are then coupled dynamically to obtain a single nonlinear model with dispersive wave propagation in finite depth and an accurate runup representation. The models are tested by runup of waves at normal incidence and comparison with a more general model for the refraction of a solitary wave on a slope. Finally, a set of runup computations for oblique solitary waves is performed and compared with estimates of oblique runup heights obtained from a combination of an analytic solution for normal incidence and optics. We find that the runup heights decrease in proportion to the square of the angle of incidence for angles up to 45°, for which the height is reduced by around 12% relative to that of normal incidence. In Appendix A, the validity of the downstream formulation is discussed in the light of solitary wave optics and wave jumps.

Radiation of short waves from the resonantly excited capillary–gravity waves

2016 ◽  
Vol 810 ◽  
pp. 5-24 ◽  
Author(s):  
M. Hirata ◽  
S. Okino ◽  
H. Hanazaki
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Gravity Waves ◽  
Wave Pattern ◽  
Short Wave ◽  
Bond Number ◽  
Linear Wave ◽  
Cnoidal Wave ◽  
Short Waves ◽  
Long Wave

Capillary–gravity waves resonantly excited by an obstacle (Froude number: $Fr=1$) are investigated by the numerical solution of the Euler equations. The radiation of short waves from the long nonlinear waves is observed when the capillary effects are weak (Bond number: $Bo<1/3$). The upstream-advancing solitary wave radiates a short linear wave whose phase velocity is equal to the solitary waves and group velocity is faster than the solitary wave (soliton radiation). Therefore, the short wave is observed upstream of the foremost solitary wave. The downstream cnoidal wave also radiates a short wave which propagates upstream in the depression region between the obstacle and the cnoidal wave. The short wave interacts with the long wave above the obstacle, and generates a second short wave which propagates downstream. These generation processes will be repeated, and the number of wavenumber components in the depression region increases with time to generate a complicated wave pattern. The upstream soliton radiation can be predicted qualitatively by the fifth-order forced Korteweg–de Vries equation, but the equation overestimates the wavelength since it is based on a long-wave approximation. At a large Bond number of $Bo=2/3$, the wave pattern has the rotation symmetry against the pattern at $Bo=0$, and the depression solitary waves propagate downstream.

The runup of solitary waves

1987 ◽  
Vol 185 ◽  
pp. 523-545 ◽  
Author(s):  
Costas Emmanuel Synolakis
Keyword(s):  
Solitary Wave ◽  
Linear Theory ◽  
Solitary Waves ◽  
Nonlinear Theory ◽  
Laboratory Experiments ◽  
Breaking Waves ◽  
Asymptotic Result ◽  
Approximate Theory ◽  
Sloping Beach

This is a study of the runup of solitary waves on plane beaches. An approximate theory is presented for non-breaking waves and an asymptotic result is derived for the maximum runup of solitary waves. A series of laboratory experiments is described to support the theory. It is shown that the linear theory predicts the maximum runup satisfactorily, and that the nonlinear theory describes the climb of solitary waves equally well. Different runup regimes are found to exist for the runup of breaking and non-breaking waves. A breaking criterion is derived for determining whether a solitary wave will break as it climbs up a sloping beach, and a different criterion is shown to apply for determining whether a wave will break during rundown. These results are used to explain some of the existing empirical runup relationships.

scholarly journals Transverse instability of interfacial solitary waves

2008 ◽  
Vol 611 ◽  
pp. 255-282 ◽  
Author(s):  
TAKESHI KATAOKA
Keyword(s):  
Solitary Wave ◽  
Asymptotic Analysis ◽  
Euler Equations ◽  
Solitary Waves ◽  
Finite Depth ◽  
Finite Amplitude ◽  
Explicit Criterion ◽  
General Statement ◽  
Transverse Instability ◽  
Long Wavelength

The linear stability of finite-amplitude interfacial solitary waves in a two-layer fluid of finite depth is examined analytically on the basis of the Euler equations. An asymptotic analysis is performed, which provides an explicit criterion of instability in the case of long-wavelength transverse disturbances. This result leads to the general statement that, when the amplitude of the solitary wave is increased, the solution becomes transversely unstable before an exchange of longitudinal stability occurs.

Resonantly interacting solitary waves

1977 ◽  
Vol 79 (1) ◽  
pp. 171-179 ◽  
Author(s):  
John W. Miles
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Rigid Wall ◽  
Resonant Interaction ◽  
Angle Of Incidence ◽  
End Points ◽  
Maximum Value ◽  
Internal Angle ◽  
Run Up ◽  
Mach Reflexion

Resonant (phase-locked) interactions among three obliquely oriented solitary waves are studied. It is shown that such interactions are associated with the parametric end points of the singular regime for interactions between two solitary waves. The latter include regular reflexion at a rigid wall, which is impossible for ϕi < (3α)½ (ϕ = angle of incidence, α = amplitude/depth [Lt ] 1), and it is shown that the observed phenomenon of ‘Mach reflexion’ can be described as a resonant interaction in this regime. The run-up at the wall is calculated as a function of ϕi/(3α)½ and is found to have a maximum value of 4αd for ϕi = (3α)½. This same resonant interaction also describes diffraction of a solitary wave at a corner of internal angle π − ψi, −(3α)½, and suggests that a solitary wave cannot turn through an angle in excess of (3α)½ at a convex corner without separating or otherwise losing its identity.

Instability of a class of dispersive solitary waves

1990 ◽  
Vol 114 (3-4) ◽  
pp. 195-212 ◽  
Author(s):  
P. E. Souganidis ◽  
W. A. Strauss
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Evolution Equations ◽  
Travelling Waves ◽  
Solitary Wave Solutions ◽  
Dispersive Waves ◽  
Weakly Nonlinear ◽  
Wave Solutions ◽  
Stability And Instability ◽  
The Stability

SynopsisThis paper studies the stability and instability properties of solitary wave solutions φ(x – ct) of a general class of evolution equations of the form Muttf(u)x=0, which support weakly nonlinear dispersive waves. It turns out that, depending on their speed c and the relation between the dispersion (i.e. the order of the pseudodifferential operator) and the nonlinearity, travelling waves maybe stable or unstable. Sharp conditions to that effect are given.

Asymptotic theory for the almost-highest solitary wave

1996 ◽  
Vol 317 ◽  
pp. 1-19 ◽  
Author(s):  
M. S. Longuet-Higgins ◽  
M. J. H. Fox
Keyword(s):  
Solitary Wave ◽  
Wave Height ◽  
Solitary Waves ◽  
Deep Water ◽  
Asymptotic Theory ◽  
Finite Depth ◽  
Periodic Waves ◽  
Wave Parameters ◽  
Asymptotic Formulae ◽  
Energy And Momentum

The behaviour of the energy in a steep solitary wave as a function of the wave height has a direct bearing on the breaking of solitary waves on a gently shoaling beach. Here it is shown that the speed, energy and momentum of a steep solitary wave in water of finite depth all behave in an oscillatory manner as functions of the wave height and as the limiting height is approached. Asymptotic formulae for these and other wave parameters are derived by means of a theory for the ‘almost-highest wave’ similar to that formulated previously for periodic waves in deep water (Longuet-Higgins & Fox 1977, 1978). It is demonstrated that the theory fits very precisely some recent calculations of solitary waves by Tanaka (1995).

On resolving fine periodic microstructures in the optical microscope

1989 ◽  
Vol 47 ◽  
pp. 364-365
Author(s):  
W.S. Putnam ◽  
C. Viney
Keyword(s):  
Liquid Crystalline ◽  
Numerical Aperture ◽  
Polarized Light ◽  
Normal Incidence ◽  
Optical Microscope ◽  
Angle Of Incidence ◽  
Resolution Limit ◽  
Crystalline Materials ◽  
Periodic Microstructures ◽  
Liquid Crystalline Materials

Many sheared liquid crystalline materials (fibers, films and moldings) exhibit a fine banded microstructure when observed in the polarized light microscope. In some cases, for example Kevlar® fiber, the periodicity is close to the resolution limit of even the highest numerical aperture objectives. The periodic microstructure reflects a non-uniform alignment of the constituent molecules, and consequently is an indication that the mechanical properties will be less than optimal. Thus it is necessary to obtain quality micrographs for characterization, which in turn requires that fine detail should contribute significantly to image formation.It is textbook knowledge that the resolution achievable with a given microscope objective (numerical aperture NA) and a given wavelength of light (λ) increases as the angle of incidence of light at the specimen surface is increased. Stated in terms of the Abbe resolution criterion, resolution improves from λ/NA to λ/2NA with increasing departure from normal incidence.

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

Solitary waves on the surface of a viscoelastic fluid running down an inclined plane

1997 ◽  
Vol 36 (2) ◽  
pp. 135-143 ◽  
Author(s):  
Bhabani Shankar Dandapat ◽  
Anadi Shankar Gupta
Keyword(s):  
Viscoelastic Fluid ◽  
Solitary Waves ◽  
Inclined Plane

Comparison principles for free-surface flows with gravity

1991 ◽  
Vol 230 ◽  
pp. 231-243 ◽  
Author(s):  
Walter Craig ◽  
Peter Sternberg
Keyword(s):  
Solitary Waves ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Immiscible Fluids ◽  
Steady Flows ◽  
Two Dimensional ◽  
Surface Flows ◽  
Ship Hulls ◽  
Geometrical Information ◽  
Supercritical Flows

This article considers certain two-dimensional, irrotational, steady flows in fluid regions of finite depth and infinite horizontal extent. Geometrical information about these flows and their singularities is obtained, using a variant of a classical comparison principle. The results are applied to three types of problems: (i) supercritical solitary waves carrying planing surfaces or surfboards, (ii) supercritical flows past ship hulls and (iii) supercritical interfacial solitary waves in systems consisting of two immiscible fluids.

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