scholarly journals RECENT DEVELOPMENTS IN THE STUDY OP BREAKING WAVES

1976 ◽  
Vol 1 (15) ◽  
pp. 24 ◽  
Author(s):  
Michael S. Longuet-Higgins
Keyword(s):  
Solitary Waves ◽  
Asymptotic Expression ◽  
Numerical Technique ◽  
Breaking Waves ◽  
Time Dependent ◽  
Wave Crest ◽  
Periodic Waves ◽  
Recent Developments ◽  
Steep Waves ◽  
Spilling Breakers

The sight and sound of breaking waves and surf is so familiar and enjoyable that we tend to forget how little we really understand about them. Why is it, that compared to other branches of wave studies our knowledge of breaking waves is so empirical and inexact? The reason must lie partly in the difficulty of finding a precise mathematical description of a fluid flow that is in general nonlinear and time-dependent. The fluid accelerations can no longer be assumed t o be small compared t o gravity, as in Stokes's theory for periodic waves and the theory of cnoidal waves in shallow water, nor is the particle velocity any longer small compared to the phase velocity. The aim of this paper is to bring together s ome recent contributions to the calculation both of steep symmetric waves and of time-dependent surface waves. These have a bearing on the behaviour of whitecaps in deep water and of surf in the breaker zone . Since spilling breakers in gently shoaling water closely resemble solitary waves, we begin with the description of solitary waves of limiting amplitude, then discuss steep waves of arbitrary height. The observed intermittency of whitecaps is discussed in terms of the energy maximum, as a function of wave steepness, In Sections 6 and 7 a simpler description of steady symmetric waves is proposed, using an asymptotic expression for the flow near the wave crest. Finally we describe a new numerical technique (MEL, or mixed Eulerian-Lagrangian) with which it has been found possible to follow the development of periodic waves past the point when overturning takes place.

scholarly journals KINEMATICS OF BREAKING WAVES

1976 ◽  
Vol 1 (15) ◽  
pp. 25 ◽  
Author(s):  
Edward B. Thornton ◽  
James J. Galvin ◽  
Frank L. Bub ◽  
David P. Richardson
Keyword(s):  
Solitary Waves ◽  
Particle Velocity ◽  
Asymptotic Expression ◽  
Breaking Waves ◽  
The Body ◽  
Time Dependent ◽  
Wave Crest ◽  
Periodic Waves ◽  
Water Particle ◽  
Highly Nonlinear

The sight and sound of breaking waves and surf is so familiar and enjoyable that we tend to forget how little we really understand about them. Why is it, that compared to other branches of wave studies our knowledge of breaking waves is so empirical and inexact? The reason must lie partly in the difficulty of finding a precise mathematical description of a fluid flow that is in general nonlinear and time-dependent. The fluid accelerations can no longer be assumed t o be small compared t o gravity, as in Stokes's theory for periodic waves and the theory of cnoidal waves in shallow water, nor is the particle velocity any longer small compared to the phase velocity. The aim of this paper is to bring together s ome recent contributions to the calculation both of steep symmetric waves and of time-dependent surface waves. These have a bearing on the behaviour of whitecaps in deep water and of surf in the breaker zone . Since spilling breakers in gently shoaling water closely resemble solitary waves, we begin with the description of solitary waves of limiting amplitude, then discuss steep waves of arbitrary height. The observed intermittency of whitecaps is discussed in terms of the energy maximum, as a function of wave steepness, In Sections 6 and 7 a simpler description of steady symmetric waves is proposed, using an asymptotic expression for the flow near the wave crest. Finally we describe a new numerical technique (MEL, or mixed Eulerian-Lagrangian) with which it has been found possible to follow the development of periodic waves past the point when overturning takes place. Measurement of waves, and vertical and horizontal water particle velocities were made of spilling, plunging and surging breakers at sandy beaches in the vicinity of Monterey, California. The measured breaking waves, derived characteristically from swell-type waves, can be described as highly nonlinear. Spectra and cross spectra were calculated between waves and velocities. Secondary waves were noted visually and by the strong harmonics in the spectra. The strength of the harmonics is related to the beach steepness, wave height and period. The phase difference between waves and horizontal velocities indicates the unstable crest of the wave leads the velocities on the average by 5-20 degrees. Phase measurements between wave gauges in a line perpendicular to the shore show breaking waves to be frequency nondispersive indicating phase-coupling of the various wave components. The coherence squared values between the sea surface elevation and the horizontal water particle velocity were high in all runs, ranging above 0.8 at the peak of the spectra. The high coherence suggests that most of the motion in the body of breaking waves is wave-induced and not turbulent.

scholarly journals RUNUP RECIPE FOR PERIODIC WAVES ON UNIFORMLY SLOPING BEACHES

1966 ◽  
Vol 1 (10) ◽  
pp. 21 ◽  
Author(s):  
Wm. G. Van Dorn
Keyword(s):  
Small Amplitude ◽  
Breaking Waves ◽  
Wave Crest ◽  
Periodic Waves ◽  
Conservation Of Energy ◽  
Wave Channel ◽  
Wave Trains ◽  
Elevation Measurement ◽  
Steep Slopes ◽  
Modified Theory

The shoaling enhancement of small-amplitude, dispersive wave trains traveling over uniform, impermeable slopes was observed in a specially-constructed wave channel, where the reproducible wave elevation measurement accuracy was about .0005-in. These observations substantially support the enhancement predicted from linear theory (conservation of energy flux) except in very shallow water and on very steep slopes, where accelerative effects become important. On the hypothesis that small-amplitude runup theory might be similarly valid for periodic waves of finite height, provided that the positive incident wave amplitude Is replaced by the local crest height above still water, this theory was modified to include the effect of the superelevation under a wave crest due to profile asymmetry. The modified theory is shown to agree acceptably with runup observations of larger waves previously reported - both for breaking and non-breaking waves. Because solutions to the modified theory cannot conveniently be obtained by manual calculation, a nomograph chart is included, from which runup predictions can be easily made, given only the wave height, period, and water depth a wavelength or so from shore, and the beach slope.

Eulerian and Lagrangian aspects of surface waves

1986 ◽  
Vol 173 ◽  
pp. 683-707 ◽  
Author(s):  
M. S. Longuet-Higgins
Keyword(s):  
Surface Waves ◽  
Gravity Waves ◽  
Breaking Waves ◽  
Capillary Waves ◽  
Wave Period ◽  
Wave Crest ◽  
Single Wave ◽  
Simple Physical Model ◽  
Vertical Accelerations ◽  
Steep Waves

Surface waves can be recorded in two kinds of ways, either with a fixed (Eulerian) probe or with a free-floating (Lagrangian) buoy. In steep waves, the differences between corresponding properties can be very marked.By a simple physical model and by accurate calculation it is shown that the Lagrangian wave period may differ from the Eulerian wave period by as much as 38 %. The Lagrangian mean level is also higher than the Eulerian mean, leading to possible discrepancies in remote sensing of the ocean from satellites.Surface accelerations are of interest in relation to the incidence of breaking waves, and for interactions between short (gravity or capillary) waves and longer gravity waves. Eulerian accelerations tend to be very non-sinusoidal, with large downwards peaks, sometimes exceeding - g in magnitude, near to sharp wave crests. Lagrangian accelerations are much smoother; for uniform gravity waves they lie between −0.388g and +0.315g. These values are verified by laboratory experiments. In wind-generated waves the limits are probably wider.In progressive gravity waves in deep water the horizontal accelerations generally exceed the vertical accelerations. In steep waves, the subsurface accelerations can slightly exceed those at the free surface.A novel application is made to the rolling motion of ships. In very steep, irrotational waves it is shown theoretically that the flow near the wave crest can lead to the rotation of the hull through angles up to 120° by a single wave, even if the wave is not breaking. This is confirmed by simple experiments. The efficiency of the keel appears to promote capsizing.

scholarly journals Bifurcation Phenomena of Nonlinear Waves in a Generalized Zakharov-Kuznetsov Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Yun Wu ◽  
Zhengrong Liu
Keyword(s):  
Solitary Waves ◽  
Nonlinear Waves ◽  
Blow Up ◽  
Periodic Waves ◽  
The Third ◽  
Bifurcation Phenomena ◽  
Kink Waves ◽  
Fourth Kind

We study the bifurcation phenomena of nonlinear waves described by a generalized Zakharov-Kuznetsov equationut+au2+bu4ux+γuxxx+δuxyy=0. We reveal four kinds of interesting bifurcation phenomena. The first kind is that the low-kink waves can be bifurcated from the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The second kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves, the symmetric solitary waves, and the 2-blow-up waves. The third kind is that the periodic-blow-up waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves.

On periodic water-waves and their convergence to solitary waves in the long-wave limit

1981 ◽  
Vol 303 (1481) ◽  
pp. 633-669 ◽  
Keyword(s):  
Integral Equation ◽  
Solitary Waves ◽  
Water Waves ◽  
Periodic Problem ◽  
Existence Theory ◽  
Periodic Waves ◽  
Equation Formulation ◽  
Long Wave ◽  
Integral Equation Formulation ◽  
Wave Limit

A detailed discussion of Nekrasov’s approach to the steady water-wave problems leads to a new integral equation formulation of the periodic problem. This development allows the adaptation of the methods of Amick & Toland (1981) to show the convergence of periodic waves to solitary waves in the long-wave limit. In addition, it is shown how the classical integral equation formulation due to Nekrasov leads, via the Maximum Principle, to new results about qualitative features of periodic waves for which there has long been a global existence theory (Krasovskii 1961, Keady & Norbury 1978).

A Single Technically Consistent Design Formula for the Thickness of Cylindrical Sections Under Internal Pressure

2006 ◽  
Vol 129 (1) ◽  
pp. 211-215 ◽  
Author(s):  
John D. Fishburn
Keyword(s):  
Experimental Data ◽  
Internal Pressure ◽  
State Of The Art ◽  
Original Data ◽  
Pressure Vessels ◽  
Time Dependent ◽  
Design Codes ◽  
Recent Developments ◽  
Current Design ◽  
Single Formula

Within the current design codes for boilers, piping, and pressure vessels, there are many different equations for the thickness of a cylindrical section under internal pressure. A reassessment of these various formulations, using the original data, is described together with more recent developments in the state of the art. A single formula, which can be demonstrated to retain the same design margin in both the time-dependent and time-independent regimes, is shown to give the best correlation with the experimental data and is proposed for consideration for inclusion in the design codes.

A UNIFIED FORMALISM OF THERMAL QUANTUM FIELD THEORY

1994 ◽  
Vol 09 (14) ◽  
pp. 2363-2409 ◽  
Author(s):  
H. CHU ◽  
H. UMEZAWA
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Vacuum Polarization ◽  
Time Dependent ◽  
Quantum Field ◽  
Bogoliubov Transformation ◽  
Transformation Matrices ◽  
Thermo Field Dynamics ◽  
The Matrix ◽  
Recent Developments

We present a comprehensive review of the most fundamental and practical aspects of thermo-field dynamics (TFD), including some of the most recent developments in the field. To make TFD fully consistent, some suitable changes in the structure of the thermal doublets and the Bogoliubov transformation matrices have been made. A close comparison between TFD and the Schwinger-Keldysh closed time path formalism (SKF) is presented. We find that TFD and SKF are in many ways the same in form; in particular, the two approaches are identical in stationary situations. However, TFD and SKF are quite different in time-dependent nonequilibrium situations. The main source of this difference is that the time evolution of the density matrix itself is ignored in SKF while in TFD it is replaced by a time-dependent Bogoliubov transformation. In this sense TFD is a better candidate for time-dependent quantum field theory. Even in equilibrium situations, TFD has some remarkable advantages over the Matsubara approach and SKF, the most notable being the Feynman diagram recipes, which we will present. We will show that the calculations of two-point functions are simplified, instead of being complicated, by the matrix nature of the formalism. We will present some explicit calculations using TFD, including space-time inhomogeneous situations and the vacuum polarization in equilibrium relativistic QED.

scholarly journals NON-BREAKING AND BREAKING WAVE LOADS ON A COOLING WATER OUTFALL

1978 ◽  
Vol 1 (16) ◽  
pp. 148
Author(s):  
G.R. Mogridge ◽  
W.W. Jamieson
Keyword(s):  
Water Level ◽  
Cooling Water ◽  
High Water ◽  
Breaking Waves ◽  
Irregular Waves ◽  
Scale Model ◽  
Breaking Wave ◽  
Wave Loads ◽  
Eastern Canada ◽  
Spilling Breakers

Cooling water from a power generating station in Eastern Canada is pumped to an outfall and distributed into the ocean through discharge ports in the sidewalls of a diffuser cap. The cap is essentially a shell-type structure consisting of a submerged circular cylinder 26.5 ft in diameter and 14 ft high. It is located in 25 ft of water at low water level and 54 ft at high water level. Horizontal forces, vertical forces and overturning moments exerted by waves on a 1:36 scale model of the diffuser cap were measured with and without cooling water discharging from the outfall. Tests were run with regular and irregular waves producing both non-breaking and breaking wave loads on the diffuser cap. The overturning moments measured on the diffuser cap were up to 150 percent greater than those on a solid submerged cylinder sealed to the seabed. Unlike sealed cylinders, all of the wave loads measured on the relatively open structure reached maximum values at approximately the same time. The largest wave loads were measured on the diffuser structure when it was subjected to spilling breakers at low water level. For a given wave height, the spilling breakers caused wave loads up to 100 percent greater than those due to non-breaking waves.

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