scholarly journals ELEMENTS OF WAVE THEORY

2010 ◽  
Vol 1 (1) ◽  
pp. 2 ◽  
Author(s):  
R. L. Wiegel ◽  
J. W. Johnson
Keyword(s):  
Surface Profile ◽  
Wave Theory ◽  
Stable Form ◽  
Periodic Waves ◽  
Arithmetical Progression ◽  
Geometrical Progression ◽  
Irrotational Motion ◽  
Opposite Manner ◽  
Surface Curve ◽  
Horizontal Bottom

The first known mathematical solution for finite height, periodic waves of stable form was developed by Gerstner (1802). From equations that were developed, Gerstner (1802) arrived at the conclusion that the surface curve was trochoidal in form. Froude (1862) and Rankine (1863) developed the theory but in the opposite manner, i.e., they started with the assumption of a trochoidal form and then developed their equations from this curve. The theory was developed for waves in water of infinite depth with the orbits of the water particles being circular, decreasing in geometrical progression as the distance below the water surface increased in arithmetical progression. Recent experiments (Wiegel, 1950) have shown that the surface profile, represented by the trochoidal equations (as well as the first few terms of Stokes' theory), closely approximates the actual profiles for waves traveling over a horizontal bottom. However the theory necessitates molecular rotation of the particles, while the manner in which waves are formed by conservative forces necessitates irrotational motion.

scholarly journals RUN-UP OF PERIODIC WAVES ON BEACHES OF NON-UNIFORM SLOPE

1984 ◽  
Vol 1 (19) ◽  
pp. 23 ◽  
Author(s):  
Yoshinobu Ogawa ◽  
Nobuo Shuto
Keyword(s):  
Experimental Data ◽  
Wave Theory ◽  
Breaking Waves ◽  
Lagrangian Description ◽  
Periodic Waves ◽  
Swash Zone ◽  
Slope Method ◽  
Long Wave ◽  
Run Up ◽  
Better Than

Run-up of periodic waves on gentle or non-uniform slopes is discussed. Breaking condition and run-up height of non-breaking waves are derived "by the use of the linear long wave theory in the Lagrangian description. As to the breaking waves, the width of swash zone and the run-up height are-obtained for relatively gentle slopes (less than 1/30), on dividing the transformation of waves into dissipation and swash processes. The formula obtained here agrees with experimental data better than Hunt's formula does. The same procedure is applied to non-uniform slopes and is found to give better results than Saville's composite slope method.

scholarly journals On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. F. R. S. &c. By Benjamin Gompertz, Esq. F. R. S

1833 ◽  
Vol 2 ◽  
pp. 252-253 ◽  
Keyword(s):  
Causes Of Death ◽  
Value Of Life ◽  
Arithmetical Progression ◽  
Short Intervals ◽  
Law Of Nature ◽  
General Law ◽  
Geometrical Progression ◽  
History Of ◽  
The Law ◽  
Life Annuities

This paper, which professes to be a continuation of former researches on the same subject printed in the Transactions of the Royal Society, is divided into two chapters. In the first the author considers the nature of the law of those numbers in tables of mortality, which express the amount of persons living at the end of ages in regular arithmetical progression. He remarks that for short intervals the law approaches nearly to a decreasing geometrical progression, and that this must be the case whatever be the strict expression for the law of mortality, provided the intervals do not exceed certain limits. But he further remarks, that this property will be found to belong to very extensive portions of tables of mortality, and instances Deparcieux’s tables, where from the age of 25 to that of 45, the numbers living at the end of each year decrease very nearly in geometrical progression. Considering however the whole extent of such a table, it will be found that the ratio of this geometrical progression is not the same in all parts of the table. But before he enters on this consideration, the author draws some consequences from the hypothesis of a geometrical progression being the strict law of nature after a certain age. One of these is the equality of value of all life annuities commencing after that age. Another is, that the want of instances in history of persons living to very enormous ages (waving those of the patriarchs,) is no proof that such may not be the law of nature, as he shows by calculation, that out of 3,000,000 persons of 92, not more than one should on this supposition reach 192. This leads him to some general considerations on the causes of death, after which he resumes the consideration of the general law of the tables.

The bifurcation of steady gravity water waves in (R, S) parameter space

1995 ◽  
Vol 302 ◽  
pp. 287-305 ◽  
Author(s):  
S. H. Doole ◽  
J. Norbury
Keyword(s):  
Parameter Space ◽  
Water Waves ◽  
Pressure Head ◽  
Wave Theory ◽  
Periodic Waves ◽  
Long Wave ◽  
S Parameter ◽  
Stokes Waves ◽  
Wave Region ◽  
Constant Period

The bifurcation of steady periodic waves from irrotational inviscid streamflows is considered. Normalizing the flux Q to unity leaves two other natural quantities R (pressure head) and S (flowforce) to parameterize the wavetrain. In a well-known paper, Benjamin & Lighthill (1954) presented calculations within a cnoidal-wave theory which suggested that the corresponding values of R and S lie inside the cusped locus traced by the sub- and supercritical streamflows. This rule has been applied since to many other flow scenarios. In this paper, regular expansions for the streamfunction and profile are constructed for a wave forming on a subcritical stream and thence values for R and S are calculated. These describe, locally, how wave brances in (R, S) parameter space point inside the streamflow cusp. Accurate numerics using a boundry-integral solver show how these constant-period branches extend globally and map out parameter space. The main result is to show that the large-amplitude branches for all steady Stokes’ waves lie surprisingly close to the subcritical stream branch, This has important consequences for the feasibility of undular bores (as opposed to hydraulic jumps) in obstructed flow. Moreover, the transition from the ‘long-wave region’ towards the ‘deep-water limit’ is char-acterized by an extreme geometry, bith of the wave branches and how they sit inside each other. It is also shown that a single (Q, R, S) trriple may represent more than one wave since the global branches can overlap in (R, S) parameter space. This non-uniqueness is not that associated with the known premature maxima of wave propertties as functions of wave amplitude near waves of greatest height.

scholarly journals The distribution of electric force in the crookes dark space

1911 ◽  
Vol 84 (573) ◽  
pp. 526-535 ◽  
Keyword(s):  
Positive Charge ◽  
Electric Force ◽  
Arithmetical Progression ◽  
Negative Glow ◽  
Geometrical Progression ◽  
Dark Space ◽  
The Subject ◽  
Metallic Wires

The electric force in the Crookes dark space and the negative glow has been the subject of a considerable number of investigations. The first determination was made by Schuster, whose results indicate the presence of a positive charge of electricity, whose density decreases in geometrical progression as the distance from the cathode increases in arithmetical progression. Graham found a curious drop in potential near the cathode, but Wehnalt, repeating these experiments, was unable to find this drop of potential, and ascribes it to the fact that the exploring points were not in the direct line of the current. Skinner§ came to the conclusion that all the fall of potential occurs at the surface of the cathode. Recently, Westphal|| has made a careful series of observations with cathodes of different metals in several gases, using a single exploring point, in which he finds a definite fall of potential— e. g . about 80 volts for Al in H 2 —at the surface of the cathode, the electric force a few millimetres away appearing from his curves nearly uniform. Now all these measurements were made by introducing exploring sounds, i. e . metallic wires or points, into the discharge, the observers trusting to these taking up the potential of the gas by which they were surrounded. The danger of such assumption has been pointed out by Sir J. J. Thomson, and for measurements made inside the dark space it seems entirely unwarranted.

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.

scholarly journals FIFTH ORDER GRAVITY WAVE THEORY

2011 ◽  
Vol 1 (7) ◽  
pp. 10 ◽  
Author(s):  
Lars Skjelbreia ◽  
James Hendrickson
Keyword(s):  
Theoretical Calculations ◽  
Wave Length ◽  
Wave Theory ◽  
Order Theory ◽  
Wave Characteristics ◽  
Finite Height ◽  
Scientists And Engineers ◽  
Fifth Order ◽  
Horizontal Bottom ◽  
Made In

In dealing with problems connected with gravity waves, scientists and engineers frequently find it necessary to make lengthy theoretical calculations involving such wave characteristics as wave height, wave length, period, and water depth. Several approximate theoretical expressions have been derived relating the above parameters. Airy, for instance, contributed a very valuable and complete theory for waves traveling over a horizontal bottom in any depth of water. Due to the simplicity of the Airy theory, it is frequently used by engineers. This theory, however, was developed for waves of very small heights and is inaccurate for waves of finite height. Stokes presented a similar solution for waves of finite height by use of trigonometric series. Using five terms in the series, this solution will extend the range covered by the Airy theory to waves of greater steepness. No attempt has been made in this paper to specify the range where the theory is applicable. The coefficients in these series are very complicated and for a numerical problem, the calculations become very tedious. Because of this difficulty, this theory would be very little used by engineers unless the value of the coefficient is presented in tabular form. The purpose of this paper is to present the results of the fifth order theory and values of the various coefficients as a function of the parameter d/L.

A presentation of cnoidal wave theory for practical application

1960 ◽  
Vol 7 (2) ◽  
pp. 273-286 ◽  
Author(s):  
R. L. Wiegel
Keyword(s):  
Particle Velocity ◽  
Wave Theory ◽  
Periodic Waves ◽  
Cnoidal Wave ◽  
Laboratory Measurements ◽  
Practical Application ◽  
Water Particle ◽  
Wave Celerity

Cnoidal wave theory is appropriate to periodic waves progressing in water whose depth is less than about one-tenth the wavelength. The leading results of existing theories are modified and given in a more practical form, and the graphs necessary to their use by engineers are presented. As well as results for the wave celerity and shape, expressions and graphs for the water particle velocity and local acceleration fields are given. A few comparisons between theory and laboratory measurements are included.

scholarly journals GENERALIZED WAVE THEORY FOR A SLOPING BED

1988 ◽  
Vol 1 (21) ◽  
pp. 12 ◽  
Author(s):  
D.H. Swart ◽  
J.B. Crowley
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Wave Theory ◽  
Breaking Waves ◽  
Surface Boundary ◽  
Data Set ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
Horizontal Bottom ◽  
Sloping Bottom

This paper discusses the development from first principles of a first-order solution for non-breaking waves on a gently sloping bottom. The theory is derived in a similar fashion as was done by Swart and Loubser (1978) for vocoidal waves on a horizontal bottom. The resulting covocoidal theory was compared to an extensive data set for waves over a sloping bottom (Nelson, 1981) and is tested for analytical validity. It adheres exactly to continuity and the kinematic free surface boundary condition, and shows comparable errors in the dynamic free surface boundary condition to that found for the better, general horizontal bed wave theories.

scholarly journals MASS TRANSPORT IN GRAVITY WAVES ON A SLOPING BOTTOM

1974 ◽  
Vol 1 (14) ◽  
pp. 25 ◽  
Author(s):  
E.W. Bijker ◽  
J.P. Th. Kalkwijk ◽  
T. Pieters
Keyword(s):  
Mass Transport ◽  
Wave Theory ◽  
Experimental Results ◽  
Wave Form ◽  
Linear Wave ◽  
Bottom Slope ◽  
Small Influence ◽  
Local Wave ◽  
Steep Slopes ◽  
Horizontal Bottom

In the present investigation the influence of bottom slope on mass transport by progressive waves was investigated, both theoretically and experimentally. Theoretical considerations based on linear wave theory show the greatest influence of the slope on the bottom drift velocities for relatively long waves and steep slopes. The numerical values, however, remain rather small (influence less than 20%). In addition, the experiments show that the bottom drift velocities are more determined by the local parameters than by the magnitude of the bottom slope m the cases examined. Considering the net bottom velocities, the discrepancy between the horizontal bottom theory (Longuet- Higgms) and experimental results is considerable. Taking into account the first harmonic of the local wave form and the small slope effect for relatively small depths in horizontal bottom theory does show, however, the same tendency as the experimental results.

Disturbance Induced by a Pressure Distribution Moving Over a Free Surface

1970 ◽  
Vol 14 (03) ◽  
pp. 195-203
Author(s):  
T. T. Huang ◽  
K. K. Wong
Keyword(s):  
Free Surface ◽  
Pressure Distribution ◽  
Surface Effect ◽  
Surface Profile ◽  
Wave Theory ◽  
Local Flow ◽  
Closed Form Solutions ◽  
Pressure Trace ◽  
Surface Effect Ship ◽  
Free Surface Profile

This paper uses the linearized water-wave theory to analyze the disturbances induced by a constant pressure distribution with a rectangular planform moving over calm water. The methods developed, however, can be applied to other pressure distributions. Numerical schemes and computation results for typical speeds and beam/length ratios are presented for the pressure trace on the sea floor when the water depth is finite and the local flow pattern when the depth is infinite. For shallow waters, closed-form solutions for both the pressure trace and free-surface profile are obtained. A surface-effect ship acts like a moving pressure distribution as far as the induced disturbances in the water are concerned. Thus, the results of the present study may be useful for the design of surface-effect ships.

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