Exact Floquet theory for waves over arbitrary periodic topographies

2012 ◽  
Vol 712 ◽  
pp. 451-470 ◽  
Author(s):  
Jie Yu ◽  
Louis N. Howard
Keyword(s):  
Analytical Form ◽  
Wave Form ◽  
Conformal Map ◽  
Flat Bottom ◽  
Linear Waves ◽  
Quantitative Results ◽  
Arbitrary Amplitude ◽  
Complete Set ◽  
Flow Domain ◽  
Wavy Topography

AbstractWe consider linear waves propagating over periodic topographies of arbitrary amplitude and wave form, generalizing the method in Howard & Yu (J. Fluid Mech., vol. 593, 2007, pp. 209–234). By a judicious construction of a conformal map from the flow domain to a uniform strip, exact solutions of Floquet type can be developed in the mapped plane. These Floquet solutions, in an essentially analytical form, are analogous to the complete set of flat-bottom propagating and evanescent waves. Therefore they can be used as a basis for the solutions of boundary value problems involving a wavy topography with a constant mean water depth. Various concrete examples are given and quantitative results are discussed. Comparisons with experimental data are made, and qualitative agreement is achieved.

scholarly journals Linear waves in two-layer fluids over periodic bottoms

2016 ◽  
Vol 794 ◽  
pp. 700-718 ◽  
Author(s):  
Jie Yu ◽  
Leo R. M. Maas
Keyword(s):  
Conformal Transformation ◽  
Floquet Theory ◽  
Analytical Form ◽  
Interfacial Wave ◽  
Flat Bottom ◽  
Surface Modes ◽  
Floquet Exponent ◽  
Long Wave ◽  
Linear Waves ◽  
Rapid Modulation

A new, exact Floquet theory is presented for linear waves in two-layer fluids over a periodic bottom of arbitrary shape and amplitude. A method of conformal transformation is adapted. The solutions are given, in essentially analytical form, for the dispersion relation between wave frequency and generalized wavenumber (Floquet exponent), and for the waveforms of free wave modes. These are the analogues of the classical Lamb’s solutions for two-layer fluids over a flat bottom. For internal modes the interfacial wave shows rapid modulation at the scale of its own wavelength that is comparable to the bottom wavelength, whereas for surface modes it becomes a long wave carrier for modulating short waves of the bottom wavelength. The approximation using a rigid lid is given. Sample calculations are shown, including the solutions that are inside the forbidden bands (i.e. Bragg resonated).

scholarly journals CURRENT DEPTH REFRACTION OF REGULAR WAVES

1984 ◽  
Vol 1 (19) ◽  
pp. 75 ◽  
Author(s):  
Ivar G. Jonsson ◽  
John B. Christoffersen
Keyword(s):  
Large Scale ◽  
Wave Motion ◽  
Regular Waves ◽  
Small Surface ◽  
First Order ◽  
Linear Waves ◽  
Sea Bed ◽  
Wave Heights ◽  
Complete Set ◽  
Bed Friction

The complete set of equations for the refraction of small surface gravity waves on large-scale currents over a gradually varying sea bed is derived and presented. Wave lengths, direction of propagation and wave heights are all determined along the so-called wave rays as solutions to ordinary, first-order differential equations. Dissipation due to bed friction in the combined current wave motion is included. The ray tracing method is used in an example. A method for the calculation of current depth refraction of weakly non-linear waves is proposed.

Explicit Analytical Solutions for a Complete Set of the Eshelby Tensors of an Ellipsoidal Inclusion

2016 ◽  
Vol 83 (12) ◽  
Author(s):  
Xiaoqing Jin ◽  
Ding Lyu ◽  
Xiangning Zhang ◽  
Qinghua Zhou ◽  
Qian Wang ◽  
...  
Keyword(s):  
Deformation Gradient ◽  
Elastic Field ◽  
Analytical Form ◽  
Ellipsoidal Inclusion ◽  
Normal Vector ◽  
Full Field ◽  
Practical Applications ◽  
Starting Point ◽  
Complete Set ◽  
Outward Unit

The celebrated solution of the Eshelby ellipsoidal inclusion has laid the cornerstone for many fundamental aspects of micromechanics. A well-known difficulty of this classical solution is to determine the elastic field outside the ellipsoidal inclusion. In this paper, we first analytically present the full displacement field of an ellipsoidal inclusion subjected to uniform eigenstrain. It is demonstrated that the displacements inside inclusion are linearly related to the coordinates and continuous across the interface of inclusion and matrix. The exterior displacement, which is less detailed in existing literatures, may be expressed in a more compact, explicit, and simpler form through utilizing the outward unit normal vector of an auxiliary confocal ellipsoid. Other than many practical applications in geological engineering, the displacement solution can be a convenient starting point to derive the deformation gradient, and subsequently in a straightforward manner to accomplish the full-field solutions of the strain and stress. Following Eshelby's definition, a complete set of the Eshelby tensors corresponding to the displacement, deformation gradient, strain, and stress are expressed in explicit analytical form. Furthermore, the jump conditions to quantify the discontinuities across the interface are discussed and a benchmark problem is provided to validate the present formulation.

scholarly journals Two-loop leading-colour QCD helicity amplitudes for two-photon plus jet production at the LHC

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Herschel A. Chawdhry ◽  
Michał Czakon ◽  
Alexander Mitov ◽  
Rene Poncelet
Keyword(s):  
Analytical Form ◽  
Hadron Colliders ◽  
Jet Production ◽  
Two Photon ◽  
Helicity Amplitudes ◽  
Complete Set

Abstract We calculate the complete set of two-loop leading-colour QCD helicity amplitudes for γγj-production at hadron colliders. Our results are presented in a compact, fully-analytical form.

scholarly journals A comparison between vertical motions measured by ADCP and inferred from temperature data

Ocean Science ◽  
2008 ◽  
Vol 4 (3) ◽  
pp. 215-222 ◽  
Author(s):  
H. van Haren
Keyword(s):  
Internal Wave ◽  
Heat Budget ◽  
Linear Part ◽  
Dominant Mode ◽  
Temperature Data ◽  
Buoyancy Frequency ◽  
Flat Bottom ◽  
Linear Mode ◽  
Linear Waves ◽  
Non Linear

Abstract. Combined vertical current (w) and thermistor string data demonstrate that high-, near-buoyancy frequency internal "wave" trains along a pycnocline in a flat-bottom shelf sea consist for 2 periods of a dominant mode-1 non-linear part, while thereafter mainly of linear [mode-2, quadrupled frequency] waves, to first order. In a simple [linear] heat budget the use of unfiltered temperature gradient or its time mean changes results by only 10%. The observations also demonstrate that temperature is not always adequate to estimate vertical motions using the linear 1-D heat equation. In shallow seas, tidal-w estimated from temperature data can be an order of magnitude weaker than directly observed w, and thus do not represent free internal waves. In the ocean, not too far from the main internal wave topography source, tidal motions represent linear waves and are well described by temperature-inferred w. There however, temperature-inferred w and directly observed w differ strongly near the buoyancy frequency, at which w is dominated by non-linear waves, and near [sub]inertial frequencies, at which w is dominated by eddies and gyroscopic waves.

CALCULATION OF ENERGY EIGENVALUES AND EIGENVECTORS FOR THREE-BODY MOLECULES IN JACOBI AND HYPERSPHERICAL COORDINATES

2010 ◽  
Vol 19 (03) ◽  
pp. 419-435 ◽  
Author(s):  
M. R. ESKANDARI ◽  
H. KHAJEHAZAD
Keyword(s):  
Center Of Mass ◽  
Analytical Form ◽  
Hyperspherical Harmonic ◽  
Mass Motion ◽  
Hyperspherical Coordinates ◽  
Energy Eigenvalues ◽  
The Matrix ◽  
Complete Set ◽  
Three Body ◽  
Differential Matrix

The Jacobi coordinates is used to eliminate center of mass motion of three-body systems. We write the results in hyperspherical coordinates and expand eigenfunction in a series of orthonormal complete set of Ykαi(Ωi) in partition i of Jacobi coordinates. The matrix elements of two-body interaction potential in hyperspherical harmonic approach are determined exactly using computed analytical form of Raynal–Revai coefficients to change the base set of Ykαi(Ωi) to other set such as Ykαj(Ωj). The generalized Laguerre functions are used to change the second order coupled differential equations to a non-differential matrix equation. This is solved to find energy eigenvalues and eigenfunctions of three-body molecules. The obtained results are in agreement with computational methods.

Arbitrary-amplitude electron acoustic solitary waves in a plasma

1995 ◽  
Vol 53 (1) ◽  
pp. 25-29 ◽  
Author(s):  
Prasanta Chatterjee ◽  
Rajkumar Roychoudhury
Keyword(s):  
Solitary Waves ◽  
Numerical Technique ◽  
Soliton Solutions ◽  
Analytical Form ◽  
Complete Agreement ◽  
Ion Temperature ◽  
Sagdeev Potential ◽  
Arbitrary Amplitude ◽  
Pseudopotential Approach ◽  
Electron Acoustic

Recently Mace et at. studied electron-acoustic solitary waves in a plasma using a pseudopotential approach. To find the finite ion-temperature Sagdeev potential, they used a numerical technique developed by Baboolal, Bharuthram & Hellberg. In this paper we show that the exact pseudopotential can be obtained in this case in an analytical form. The numerical results obtained by Mace et at. are compared with our result, and complete agreement is found. We also discuss the conditions for the existence of solitary-wave solutions, and obtain the soliton solutions in some cases when these conditions are satisfied.

Edge waves on a gently sloping beach

1989 ◽  
Vol 199 ◽  
pp. 125-131 ◽  
Author(s):  
John Miles
Keyword(s):  
Free Surface ◽  
Turning Point ◽  
Surface Displacement ◽  
Dominant Mode ◽  
Variational Approximation ◽  
Edge Waves ◽  
Flat Bottom ◽  
Free Surface Displacement ◽  
Complete Set ◽  
Sloping Beach

Edge waves of frequency ω and longshore wavenumber k in water of depth h(y) = h1H(σy/h1), 0 [les ] y < ∞, are calculated through an asymptotic expansion in σ/kh1 on the assumptions that σ [Lt ] 1 and kh1 = O(1). Approximations to the free-surface displacement in an inner domain that includes the singular point at h = 0 and the turning point near gh ≈ ω2/K2 and to the eigenvalue λ ≡ ω2/σgh are obtained for the complete set of modes on the assumption that h(y) is analytic. A uniformly valid approximation for the free-surface displacement and a variational approximation to Λ are obtained for the dominant mode. The results are compared with the shallow-water approximations of Ball (1967) for a slope that decays exponentially from σ to 0 as h increases from 0 to h1 and of Minzoni (1976) for a uniform slope that joins h = 0 to a flat bottom at h = h1 and with the geometrical-optics approximation of Shen, Meyer & Keller (1968).

scholarly journals A comparison between vertical motions measured by ADCP and inferred from temperature data

2008 ◽  
Vol 5 (1) ◽  
pp. 103-121
Author(s):  
H. van Haren
Keyword(s):  
Internal Wave ◽  
Heat Budget ◽  
Linear Part ◽  
Dominant Mode ◽  
Temperature Data ◽  
Buoyancy Frequency ◽  
Flat Bottom ◽  
Linear Mode ◽  
Linear Waves ◽  
Non Linear

Abstract. Combined vertical current (w) and thermistor string data demonstrate that high-frequency internal "wave" trains along a pycnocline in a flat-bottom shelf sea consist for 2 periods of a dominant mode-1 non-linear part, while thereafter mainly of linear [mode-2, quadrupled frequency] waves, to first order. In a simple [linear] heat budget, the use of unfiltered temperature gradient or its time mean changes results by only 10%. The observations also demonstrate that temperature is not always adequate to estimate vertical motions using the linear 1-D heat equation. In shallow seas, tidal-w estimated from temperature data can be an order of magnitude weaker than directly observed w, and thus do not represent free internal waves. In the open ocean, tidal motions represent linear waves and are well described by temperature-inferred w. There however, the internal wave continuum is not well-described: near the buoyancy frequency it is dominated by non-linear waves and near [sub]inertial frequencies by eddies and gyroscopic waves.

Trends in Electron Energy Loss Spectroscopy

1977 ◽  
Vol 35 ◽  
pp. 228-229
Author(s):  
C. Colliex ◽  
P. Trebbia
Keyword(s):  
Energy Loss ◽  
Electron Energy ◽  
Electron Energy Loss Spectroscopy ◽  
Materials Science ◽  
Analytical Technique ◽  
Recent Developments ◽  
Quantitative Results ◽  
Electron Microscopes ◽  
Electron Energy Loss ◽  
Primary Electrons

The physical foundations for the use of electron energy loss spectroscopy towards analytical purposes, seem now rather well established and have been extensively discussed through recent publications. In this brief review we intend only to mention most recent developments in this field, which became available to our knowledge. We derive also some lines of discussion to define more clearly the limits of this analytical technique in materials science problems.The spectral information carried in both low ( 0<ΔE<100eV ) and high ( >100eV ) energy regions of the loss spectrum, is capable to provide quantitative results. Spectrometers have therefore been designed to work with all kinds of electron microscopes and to cover large energy ranges for the detection of inelastically scattered electrons (for instance the L-edge of molybdenum at 2500eV has been measured by van Zuylen with primary electrons of 80 kV). It is rather easy to fix a post-specimen magnetic optics on a STEM, but Crewe has recently underlined that great care should be devoted to optimize the collecting power and the energy resolution of the whole system.

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