A variational method for love waves in nonhorizontally layered structures

1973 ◽  
Vol 63 (3) ◽  
pp. 1013-1023
Author(s):  
B. Gjevik
Keyword(s):  
Travel Time ◽  
Wave Train ◽  
Love Waves ◽  
Wave Number ◽  
Transient Wave ◽  
Varying Thickness ◽  
Wave Trains ◽  
The Mean ◽  
Gradual Variation ◽  
Impulsive Source

abstract A variational principle for Love waves propagating in a layer with varying thickness overlying a half-space is formulated. By an averaging procedure, the wave modulation due to gradual variation of the thickness of the upper layer is found. The case of long-period Love waves is particularly examined and the wave number and amplitude modulation both for a monochromatic wave and for a transient wave train due to an impulsive source are determined. Moreover, the travel time for a certain phase in the wave train is determined. It is found that for well-dispersed wave trains the travel time is given by the phase velocity corresponding to the mean depth of the upper layer.

On the mean motion induced by internal gravity waves

1969 ◽  
Vol 36 (4) ◽  
pp. 785-803 ◽  
Author(s):  
Francis P. Bretherton
Keyword(s):  
Gravity Waves ◽  
Wave Energy ◽  
Wave Train ◽  
Three Dimensional ◽  
Internal Gravity Waves ◽  
Wave Number ◽  
Mean Flow ◽  
Mean Velocity ◽  
Mean Motion ◽  
The Mean

A train of internal gravity waves in a stratified liquid exerts a stress on the liquid and induces changes in the mean motion of second order in the wave amplitude. In those circumstances in which the concept of a slowly varying quasi-sinusoidal wave train is consistent, the mean velocity is almost horizontal and is determined to a first approximation irrespective of the vertical forces exerted by the waves. The sum of the mean flow kinetic energy and the wave energy is then conserved. The circulation around a horizontal circuit moving with the mean velocity is increased in the presence of waves according to a simple formula. The flow pattern is obtained around two- and three-dimensional wave packets propagating into a liquid at rest and the results are generalized for any basic state of motion in which the internal Froude number is small. Momentum can be associated with a wave packet equal to the horizontal wave-number times the wave energy divided by the intrinsic frequency.

Non-linear dispersion of water waves

1967 ◽  
Vol 27 (2) ◽  
pp. 399-412 ◽  
Author(s):  
G. B. Whitham
Keyword(s):  
Water Waves ◽  
Wave Train ◽  
Periodic Wave ◽  
Wave Number ◽  
Amplitude Wave ◽  
Linear Dispersion ◽  
Elliptic Case ◽  
Non Linear ◽  
Wave Trains ◽  
Linear Water Waves

The slow dispersion of non-linear water waves is studied by the general theory developed in an earlier paper (Whitham 1965b). The average Lagrangian is calculated from the Stokes expansion for periodic wave trains in water of arbitrary depth. This Lagrangian can be used for the various applications described in the above reference. In this paper, the crucial question of the ‘type’ of the differential equations for the wave-train parameters (local amplitude, wave-number, etc.) is established. The equations are hyperbolic or elliptic according to whetherkh0is less than or greater than 1.36, wherekis the wave-number per 2π andh0is the undisturbed depth. In the hyperbolic case, changes in the wave train propagate and the characteristic velocities give generalizations of the linear group velocity. In the elliptic case, modulations in the wave train grow exponentially and a periodic wave train will be unstable in this sense; thus, periodic wave trains on water will be unstable ifkh0> 1·36, The instability of deep-water waves,kh0> 1·36, was discovered in a different way by Benjamin (1966). The relation between the two approaches is explained.

Phase velocity effects in tertiary wave interactions

1962 ◽  
Vol 12 (3) ◽  
pp. 333-336 ◽  
Author(s):  
M. S. Longuet-Higgins ◽  
O. M. Phillips
Keyword(s):  
Phase Velocity ◽  
Continuous Spectrum ◽  
Deep Water ◽  
Wave Train ◽  
The Other ◽  
Wave Number ◽  
Wave Interactions ◽  
Single Wave ◽  
Phase Velocities ◽  
Wave Trains

It is shown that, when two trains of waves in deep water interact, the phase velocity of each is modified by the presence of the other. The change in phase velocity is of second order and is distinct from the increase predicted by Stokes for a single wave train. When the wave trains are moving in the same direction, the increase in velocity Δc2 of the wave with amplitude a2, wave-number k2 and frequency α2 resulting from the interaction with the wave (a1, k1, σ1) is given by Δc2 = a21k1σ1, provided k1 < k2. If k1 > k2, then Δc2 is given by the same expression multiplied by k2/k1. If the directions of propagation are opposed, the phase velocities are decreased by the same amount. These expressions are extended to give the increase (or decrease) in velocity due to a continuous spectrum of waves all travelling in the same (or opposite) direction.

The disintegration of wave trains on deep water Part 1. Theory

1967 ◽  
Vol 27 (3) ◽  
pp. 417-430 ◽  
Author(s):  
T. Brooke Benjamin ◽  
J. E. Feir
Keyword(s):  
Deep Water ◽  
Wave Train ◽  
Low Frequency ◽  
Periodic Wave ◽  
Wave Number ◽  
Fundamental Wave ◽  
Side Band ◽  
Wave Trains ◽  
The Stability ◽  
Deep Water Part

The phenomenon in question arises when a periodic progressive wave train with fundamental frequency ω is formed on deep water—say by radiation from an oscillating paddle—and there are also present residual wave motions at adjacent side-band frequencies ω(1 ± δ), such as would be generated if the movement of the paddle suffered a slight modulation at low frequency. In consequence of coupling through the non-linear boundary conditions at the free surface, energy is then transferred from the primary motion to the side bands at a rate that, as will be shown herein, can increase exponentially as the interaction proceeds. The result is that the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start.In this paper a theoretical investigation is made into the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes, and Part 2 which will follow is an account of some experimental observations in accord with the present predictions. The main conclusion of the theory is that infinitesimal disturbances of the type considered will undergo unbounded magnification if \[ 0 < \delta \leqslant (\sqrt{2})ka, \] where k and a are the fundamental wave-number and amplitude of the perturbed wave train. The asymptotic rate of growth is a maximum for δ = ka.

Phase-matched filters: Application to the study of Love waves

1979 ◽  
Vol 69 (1) ◽  
pp. 27-44
Author(s):  
Tom Goforth ◽  
Eugene Herrin
Keyword(s):  
Rayleigh Waves ◽  
Matched Filter ◽  
Multipath Propagation ◽  
Group Velocity Dispersion ◽  
Wave Train ◽  
Seismic Signal ◽  
Love Waves ◽  
Complex Spectrum ◽  
Matched Filters ◽  
Wave Trains

abstract Seismic surface waves are usually composed of overlapping wave trains representing multipath propagation. A first task in the analysis of such waves is to identify and separate the various component wave trains so that each can be analyzed separately. Phase-matched filters are a class of linear filters in which the Fourier phase of the filter is made equal to that of a given signal. Herrin and Goforth (1977) described an iterative technique which can be used to find a phase-matched filter for a particular component of a seismic signal. Application of the filters to digital records of Rayleigh waves allowed multiple arrivals to be identified and removed, and allowed recovery of the complex spectrum of the primary wave train along with its apparent group-velocity dispersion curve. A comparable analysis of Love waves presents additional complications. Love waves are contaminated by both Love and Rayleigh multipathing and by primary off-axis Rayleigh energy. In the case of explosions, there is much less energy generated as Love waves than as Rayleigh waves. The applicability of phase-matched filtering to Love waves is demonstrated by its use on earthquakes occurring in the Norwegian Sea and near Iceland and on a nuclear explosion in Novaya Zemlya. Despite severe multipathing in two of the three events, the amplitude and phase of each of the primary Love waves were recovered without significant distortion.

scholarly journals Investigation of dominant traveling 10-day wave components using long-term MERRA-2 database

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Chunming Huang ◽  
Wei Li ◽  
Shaodong Zhang ◽  
Gang Chen ◽  
Kaiming Huang ◽  
...  
Keyword(s):  
Flow Instability ◽  
Transmission Condition ◽  
Wave Number ◽  
Mean Flow ◽  
Excited Wave ◽  
Propagating Waves ◽  
Zonal Wave Number ◽  
The Mean

AbstractThe eastward- and westward-traveling 10-day waves with zonal wavenumbers up to 6 from surface to the middle mesosphere during the recent 12 years from 2007 to 2018 are deduced from MERRA-2 data. On the basis of climatology study, the westward-propagating wave with zonal wave number 1 (W1) and eastward-propagating waves with zonal wave numbers 1 (E1) and 2 (E2) are identified as the dominant traveling ones. They are all active at mid- and high-latitudes above the troposphere and display notable month-to-month variations. The W1 and E2 waves are strong in the NH from December to March and in the SH from June to October, respectively, while the E1 wave is active in the SH from August to October and also in the NH from December to February. Further case study on E1 and E2 waves shows that their latitude–altitude structures are dependent on the transmission condition of the background atmosphere. The presence of these two waves in the stratosphere and mesosphere might have originated from the downward-propagating wave excited in the mesosphere by the mean flow instability, the upward-propagating wave from the troposphere, and/or in situ excited wave in the stratosphere. The two eastward waves can exert strong zonal forcing on the mean flow in the stratosphere and mesosphere in specific periods. Compared with E2 wave, the dramatic forcing from the E1 waves is located in the poleward regions.

Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell

1969 ◽  
Vol 39 (3) ◽  
pp. 477-495 ◽  
Author(s):  
R. A. Wooding
Keyword(s):  
Porous Medium ◽  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Saturated Porous Medium ◽  
Mean Amplitude ◽  
Wave Number ◽  
Two Fluids ◽  
Averaged Equations ◽  
The Mean ◽  
Hele Shaw Cell

Waves at an unstable horizontal interface between two fluids moving vertically through a saturated porous medium are observed to grow rapidly to become fingers (i.e. the amplitude greatly exceeds the wavelength). For a diffusing interface, in experiments using a Hele-Shaw cell, the mean amplitude taken over many fingers grows approximately as (time)2, followed by a transition to a growth proportional to time. Correspondingly, the mean wave-number decreases approximately as (time)−½. Because of the rapid increase in amplitude, longitudinal dispersion ultimately becomes negligible relative to wave growth. To represent the observed quantities at large time, the transport equation is suitably weighted and averaged over the horizontal plane. Hyperbolic equations result, and the ascending and descending zones containing the fronts of the fingers are replaced by discontinuities. These averaged equations form an unclosed set, but closure is achieved by assuming a law for the mean wave-number based on similarity. It is found that the mean amplitude is fairly insensitive to changes in wave-number. Numerical solutions of the averaged equations give more detailed information about the growth behaviour, in excellent agreement with the similarity results and with the Hele-Shaw experiments.

Crustal structure in the Transvaal*

1956 ◽  
Vol 46 (4) ◽  
pp. 293-316
Author(s):  
P. G. Gane ◽  
A. R. Atkins ◽  
J. P. F. Sellschop ◽  
P. Seligman
Keyword(s):  
Travel Time ◽  
Crustal Structure ◽  
Time Data ◽  
Lower Velocity ◽  
Travel Time Data ◽  
The Mean

abstract Travel-time data are given at 25 km. intervals between 50 and 500 km. for traverses west, south, east, and north of Johannesburg. These derive from numerous seismograms of Witwatersrand earth tremors taken by means of a triggering technique. The only phases considered to be consistent are those mentioned below, and few signs of a change of velocity with depth were discovered. There were no great differences in the results for the various directions, and the mean results were: P 1 = + 0.24 + Δ / 6.18 sec . S 1 = + 0.37 + Δ / 3.66 sec . P n = + 7.61 + Δ / 8.27 sec . S n = + 11.4 + Δ / 4.73 sec . which give crustal depths of 35.1 and 33.3 km. from P and S data respectively. These depths include about 1.3 km. of superficial material of lower velocity.

Analytical approach unifying the two-regime scaling for bedload particle motions

2021 ◽  
Author(s):  
Zi Wu ◽  
Arvind Singh ◽  
Efi Foufoula-Georgiou ◽  
Michele Guala ◽  
Xudong Fu ◽  
...  
Keyword(s):  
Travel Time ◽  
Velocity Variation ◽  
Travel Times ◽  
Scaling Exponents ◽  
Time Scaling ◽  
Dispersion Regime ◽  
The Mean ◽  
Bedload Sediment ◽  
Distance Travel ◽  
First Time

&lt;p&gt;Bedload particle hops are defined as successive motions of a particle from start to stop, characterizing one of the most fundamental processes describing bedload sediment transport in rivers. Although two transport regimes have been recently identified for short- and long-hops, respectively &lt;strong&gt;(Wu et al., &lt;em&gt;Water Resour Res&lt;/em&gt;, 2020)&lt;/strong&gt;, there still lacks a theory explaining how the mean hop distance-travel time scaling may extend to cover the phenomenology of bedload particle motions. Here we propose a velocity-variation based formulation, and for the first time, we obtain analytical solution for the mean hop distance-travel time relation valid for the entire range of travel times, which agrees well with the measured data &lt;strong&gt;(Wu et al., &lt;em&gt;J Fluid Mech&lt;/em&gt;, 2021)&lt;/strong&gt;. Regarding travel times, we identify three distinct regimes in terms of different scaling exponents: respectively as ~1.5 for an initial regime and ~5/3 for a transition regime, which define the short-hops; and 1 for the so-called Taylor dispersion regime defining long-hops. The corresponding probability density function of the hop distance is also analytically obtained and experimentally verified.&amp;#160;&lt;/p&gt;

Stability of periodic waves in shallow water

1974 ◽  
Vol 66 (1) ◽  
pp. 81-96 ◽  
Author(s):  
P. J. Bryant
Keyword(s):  
Shallow Water ◽  
Wave Train ◽  
Periodic Wave ◽  
Fundamental Wave ◽  
Cnoidal Waves ◽  
Margin Of Stability ◽  
Fundamental Wavelength ◽  
Stokes Waves ◽  
Wave Trains ◽  
Permanent Shape

Waves of small but finite amplitude in shallow water can occur as periodic wave trains of permanent shape in two known forms, either as Stokes waves for the shorter wavelengths or as cnoidal waves for the longer wavelengths. Calculations are made here of the periodic wave trains of permanent shape which span uniformly the range of increasing wavelength from Stokes waves to cnoidal waves and beyond. The present investigation is concerned with the stability of such permanent waves to periodic disturbances of greater or equal wavelength travelling in the same direction. The waves are found to be stable to infinitesimal and to small but finite disturbances of wavelength greater than the fundamental, the margin of stability decreasing either as the fundamental wave becomes more nonlinear (i.e. contains more harmonics), or as the wavelength of the periodic disturbance becomes large compared with the fundamental wavelength. The decreasing margin of stability is associated with an increasing loss of spatial periodicity of the wave train, to the extent that small but finite disturbances can cause a form of interaction between consecutive crests of the disturbed wave train. In such a case, a small but finite disturbance of wavelength n times the fundamental wavelength converts the wave train into n interacting wave trains. The amplitude of the disturbance subharmonic is then nearly periodic, the time scale being the time taken for repetitions of the pattern of interactions. When the disturbance is of the same wavelength as the permanent wave, the wave is found to be neutrally stable both to infinitesimal and to small but finite disturbances.

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