scholarly journals The asymptotic theory of dispersion relations containing Bessel functions of imaginary order

2012 ◽  
Vol 468 (2148) ◽  
pp. 4008-4023 ◽  
Author(s):  
C. J. Chapman
Keyword(s):  
Dispersion Relation ◽  
Asymptotic Theory ◽  
Bessel Functions ◽  
Removable Singularity ◽  
Wave Theory ◽  
Dispersion Relations ◽  
Dynamical Problem ◽  
Physical Processes ◽  
Analytical Approximations ◽  
Wide Range

This paper presents a method of analysing wave-field dispersion relations in which Bessel functions of imaginary order occur. Such dispersion relations arise in applied studies in oceanography and astronomy, for example. The method involves the asymptotic theory developed by Dunster in 1990, and leads to simple analytical approximations containing only trigonometric and exponential functions. Comparisons with accurate numerical calculations show that the resulting approximations to the dispersion relation are highly accurate. In particular, the approximations are powerful enough to reveal the fine structure in the dispersion relation and so identify different wave regimes corresponding to different balances of physical processes. Details of the method are presented for the fluid-dynamical problem that stimulated this analysis, namely the dynamics of an internal ocean wave in the presence of an aerated surface layer; the method identifies and gives different approximations for the subcritical, supercritical and critical regimes. The method is potentially useful in a wide range of problems in wave theory and stability theory. A mathematical theme of the paper is that of the removable singularity.

The finite-product method in the theory of waves and stability

2009 ◽  
Vol 466 (2114) ◽  
pp. 471-491 ◽  
Author(s):  
C. J. Chapman ◽  
S. V. Sorokin
Keyword(s):  
Dispersion Relation ◽  
Elastic Waves ◽  
Low Frequency ◽  
Wave Theory ◽  
Finite Product ◽  
High Frequencies ◽  
Polynomial Approximations ◽  
Wide Range ◽  
Product Method ◽  
Finite Products

This paper presents a method of analysing the dispersion relation and field shape of any type of wave field for which the dispersion relation is transcendental. The method involves replacing each transcendental term in the dispersion relation by a finite-product polynomial. The finite products chosen must be consistent with the low-frequency, low-wavenumber limit; but the method is nevertheless accurate up to high frequencies and high wavenumbers. Full details of the method are presented for a non-trivial example, that of anti-symmetric elastic waves in a layer; the method gives a sequence of polynomial approximations to the dispersion relation of extraordinary accuracy over an enormous range of frequencies and wavenumbers. It is proved that the method is accurate because certain gamma-function expressions, which occur as ratios of transcendental terms to finite products, largely cancel out, nullifying Runge’s phenomenon. The polynomial approximations, which are unrelated to Taylor series, introduce no spurious branches into the dispersion relation, and are ideal for numerical computation. The method is potentially useful for a very wide range of problems in wave theory and stability theory.

scholarly journals The deferred limit method for long waves in a curved waveguide

2017 ◽  
Vol 473 (2200) ◽  
pp. 20160900 ◽  
Author(s):  
C. J. Chapman ◽  
S. V. Sorokin
Keyword(s):  
Dispersion Relation ◽  
Elastic Waves ◽  
Bessel Functions ◽  
Dispersion Relations ◽  
High Order ◽  
Long Waves ◽  
Strongly Coupled ◽  
Curved Waveguide

This paper presents a technique, based on a deferred approach to a limit, for analysing the dispersion relation for propagation of long waves in a curved waveguide. The technique involves the concept of an analytically satisfactory pair of Bessel functions, which is different from the concept of a numerically satisfactory pair, and simplifies the dispersion relations for curved waveguide problems. Details are presented for long elastic waves in a curved layer, for which symmetric and antisymmetric waves are strongly coupled. The technique gives high-order corrections to a widely used approximate dispersion relation based a kinematic hypothesis, and determines rigorously which of its coefficients are exact.

Magnetic Superlattice: Localized Magnetostatic Waves and Magnetic Polaritons

2003 ◽  
Vol 17 (15) ◽  
pp. 829-839
Author(s):  
R. T. Tagiyeva ◽  
M. Saglam
Keyword(s):  
Magnetic Material ◽  
Electromagnetic Wave ◽  
Dispersion Relation ◽  
Wave Theory ◽  
Frequency Region ◽  
Long Wavelength ◽  
Magnetostatic Waves ◽  
General Dispersion ◽  
Magnetic Polaritons ◽  
Magnetic Superlattice

Localized magnetostatic waves and magnetic polaritons at the junction of the magnetic material and magnetic superlattice composed of the alternating ferromagnetic or ferromagnetic and nonmagnetic layers are investigated in the framework of the electromagnetic wave theory in Voigt geometry. The general dispersion relation for localized magnetic polaritons and magnetostatic waves (MW) are derived in the long-wavelength limit. The dispersion curves and frequency region of the exsistence of the localized MW and magnetic polaritons are calculated numerically.

Dispersion relations of dust lattice waves in two-dimensional honeycomb configuration

2013 ◽  
Vol 79 (5) ◽  
pp. 629-633
Author(s):  
B. FAROKHI
Keyword(s):  
Dispersion Relation ◽  
Group Velocity ◽  
Hexagonal Lattice ◽  
Dispersion Relations ◽  
Two Dimensional ◽  
The Third ◽  
Lattice Waves ◽  
The Waves

AbstractThe linear dust lattice waves propagating in a two-dimensional honeycomb configuration is investigated. The interaction between particles is considered up to distance 2a, i.e. the third-neighbor interactions. Longitudinal and transverse (in-plane) dispersion relations are derived for waves in arbitrary directions. The study of dispersion relations with more neighbor interactions shows that in some cases the results change physically. Also, the dispersion relation in the different direction displays anisotropy of the group velocity in the lattice. The results are compared with dispersion relations of the waves in the hexagonal lattice.

scholarly journals Spontaneous and forced decay of atomic nuclei realized in two stages

2020 ◽  
pp. 122-135
Author(s):  
Viacheslav S. Okunev
Keyword(s):  
Stable Isotopes ◽  
Kinetic Energy ◽  
Nuclear Reactions ◽  
External Influence ◽  
Physical Processes ◽  
Atomic Nuclei ◽  
Wide Range ◽  
Fragmentation Reactions ◽  
Two Stages ◽  
Principle Of Similarity

The main purpose of the work is to determine the possibility of cluster decays of superheavy atomic nuclei. The universality of the principle of similarity allows you to apply it to the analysis of not studied physical processes. Analogies are observed in forced and spontaneous decays of atomic nuclei. It is shown that in two stages, processes initiated by external influence are realized: fragmentation reactions, forced fission of stable nuclei, and impact radioactivity. Nuclear reactions of fragmentation and forced fission of stable isotopes of lead and bismuth are realized under the action of particles (hadrons) and light atomic nuclei with a kinetic energy of more than 108 eV. Shock radioactivity is observed in the collision of macroobjects having a crystalline structure at speeds of at least ∼1 km/s. Also, in two stages, some radioactive decays of atomic nuclei are realized, including extremely rare cluster decays. Based on the analogies of the processes considered, some cautious predictions are made about the possibility of cluster decays of atomic nuclei in a wide range of atomic masses.

scholarly journals Life, Birth and Death in Democritus. Atomistic Reflections Between Physics and Ethics

2017 ◽  
Vol 8 (1) ◽  
pp. 141-154
Author(s):  
Miriam Campolina Diniz Peixoto
Keyword(s):  
Human Behaviour ◽  
Physical Processes ◽  
Wide Range ◽  
The Subject ◽  
Birth Death

The subject of life, birth and death constitutes one of the main topics in Democritus’ reflection on human questions. He seeks to understand what men think about the processes of birth and death and how they, accordingly, determine their behavior and attitudes. His reflections comprise a wide range of perspectives and aspects that include examining human behaviour and investigating how it reveals a certain temperament or inclination, inquiring about the nature of these processes and extending the analyses of the processes of birth and death to whole beings through the couple generation-corruption. In the present paper, I intend to examine the main theses and arguments which appear in the testimonies and fragments through which Democritus’ thought was transmitted from antiquity. Furthermore, I will also discuss the hypotheses that for Democritus the most important opposition was not life-death, but rather birth-death and that, at the same time, his idea of nature and life comprises both processes in the perspective of atomistic philosophy. I shall show that corruption has to be considered in two different ways, that is, in the context of physical processes that keep the kosmos in its persistence and in the context of the existence of natural beings, both living and lifeless.

scholarly journals The fall and rise of consciousness science

2018 ◽  
Author(s):  
Anil Seth
Keyword(s):  
Twentieth Century ◽  
Empirical Study ◽  
Neurological Disorders ◽  
Neural Mechanisms ◽  
Central Feature ◽  
Hard Problem ◽  
Physical Processes ◽  
Wide Range ◽  
Third Person ◽  
The Universe

At the birth of psychology as a science, consciousness was its central problem. But throughout the twentieth century, ideological and methodological concerns pushed the explicit empirical study of consciousness to the sidelines. Since the 1990s, studying consciousness has regained a legitimacy and impetus befitting its status as the central feature of our mental lives. Nowadays consciousness science encompasses a rich interdisciplinary mixture drawing together philosophical, theoretical, computational, experimental, and clinical perspectives. While solving the metaphysically ‘hard’ problem of why consciousness is part of the universe may seem as intractable as ever, scientists have learned a great deal about the neural mechanisms underlying conscious states. Further progress will depend on specifying closer explanatory mappings between (first person subjective) phenomenological descriptions and (third person objective) descriptions of biological and physical processes. Such progress will help reframe our understanding of our place in nature, and may also accelerate clinical approaches to a wide range of psychiatric and neurological disorders.

scholarly journals Physical Transport Processes that Affect the Distribution of Oil in the Gulf of Mexico: Observations and Modeling

Oceanography ◽  
2021 ◽  
Vol 34 (1) ◽  
pp. 58-75
Author(s):  
Michel Boufadel ◽  
◽  
Annalisa Bracco ◽  
Eric Chassignet ◽  
Shuyi Chen ◽  
...  
Keyword(s):  
Gulf Of Mexico ◽  
Physical Environment ◽  
Large Scale ◽  
Transport Processes ◽  
Small Scale ◽  
Surface Mixed Layer ◽  
Physical Processes ◽  
Mixing Processes ◽  
Near Surface ◽  
Wide Range

Physical transport processes such as the circulation and mixing of waters largely determine the spatial distribution of materials in the ocean. They also establish the physical environment within which biogeochemical and other processes transform materials, including naturally occurring nutrients and human-made contaminants that may sustain or harm the region’s living resources. Thus, understanding and modeling the transport and distribution of materials provides a crucial substrate for determining the effects of biological, geological, and chemical processes. The wide range of scales in which these physical processes operate includes microscale droplets and bubbles; small-scale turbulence in buoyant plumes and the near-surface “mixed” layer; submesoscale fronts, convergent and divergent flows, and small eddies; larger mesoscale quasi-geostrophic eddies; and the overall large-scale circulation of the Gulf of Mexico and its interaction with the Atlantic Ocean and the Caribbean Sea; along with air-sea interaction on longer timescales. The circulation and mixing processes that operate near the Gulf of Mexico coasts, where most human activities occur, are strongly affected by wind- and river-induced currents and are further modified by the area’s complex topography. Gulf of Mexico physical processes are also characterized by strong linkages between coastal/shelf and deeper offshore waters that determine connectivity to the basin’s interior. This physical connectivity influences the transport of materials among different coastal areas within the Gulf of Mexico and can extend to adjacent basins. Major advances enabled by the Gulf of Mexico Research Initiative in the observation, understanding, and modeling of all of these aspects of the Gulf’s physical environment are summarized in this article, and key priorities for future work are also identified.

scholarly journals Originals of operating images for generalized problems of unsteady heat conductivity

2019 ◽  
Vol 14 (4) ◽  
pp. 77-86 ◽  
Author(s):  
E. M. Kartashov
Keyword(s):  
Mathematical Models ◽  
Bessel Functions ◽  
Applied Mathematics ◽  
Operational Calculus ◽  
Polymeric Materials ◽  
Thermal Reaction ◽  
Heaviside Step Function ◽  
External Influences ◽  
Wide Range ◽  
Variable Function

A series of operating (Laplace) non-standard images, the originals of which are absent in well-known reference books on operational calculus, are considered. By reducing one of the basic images to the Riemann-Mellin contour integral for the modified Bessel functions and analyzing the corresponding inversion formula using the approaches of the complex variable function theory, an analytical form of the original original is found, which is abrupt in nature with a break point. It is shown that analytical solutions of the corresponding mathematical models using the found originals have a wave character, which is expressed by the presence of the Heaviside step function in the solutions. The latter means that at any time there is a region of physical disturbance to the point of discontinuity and an unperturbed area after the point of discontinuity. The images studied are included in the operational solutions of mathematical models in many areas of applied mathematics. physics, thermomechanics, thermal physics, in particular in the theory of thermal shock of viscoelastic bodies, in the study of the thermal reaction of solids based on the classical Maxwell-Cattaneo-Lykov-Vernott phenomenology, taking into account the final rate of heat propagation. These models are needed to study the thermal reaction of relatively new consolidated structurally sensitive polymeric materials in structures exposed to high-intensity external influences. The analytical relations obtained for the originals and the original improper integrals resulting from them, containing combinations of Bessel functions, can be used in the general methodology of constructing and applying various mathematical models in a wide range of external influences on materials in many fields of science and technology.

Ion-cyclotron modes in weakly relativistic plasmas

1994 ◽  
Vol 51 (3) ◽  
pp. 371-379 ◽  
Author(s):  
Chandu Venugopal ◽  
P. J. Kurian ◽  
G. Renuka
Keyword(s):  
Dispersion Relation ◽  
High Frequency ◽  
Low Frequency ◽  
Dispersion Relations ◽  
The Other ◽  
Larmor Radius ◽  
Relativistic Plasmas ◽  
Ion Plasma ◽  
Maxwellian Plasma ◽  
Relativistic Dispersion

We derive a dispersion relation for the perpendicular propagation of ioncyclotron waves around the ion gyrofrequency ω+ in a weaklu relaticistic anisotropic Maxwellian plasma. These waves, with wavelength greater than the ion Larmor radius rL+ (k⊥ rL+ < 1), propagate in a plasma characterized by large ion plasma frequencies (). Using an ordering parameter ε, we separated out two dispersion relations, one of which is independent of the relativistic terms, while the other depends sensitively on them. The solutions of the former dispersion relation yield two modes: a low-frequency (LF) mode with a frequency ω < ω+ and a high-frequency (HF) mode with ω > ω+. The plasma is stable to the propagation of these modes. The latter dispersion relation yields a new LF mode in addition to the modes supported by the non-relativistic dispersion relation. The two LF modes can coalesce to make the plasma unstable. These results are also verified numerically using a standard root solver.

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