SOLUTIONS OF THE HELMHOLTZ EQUATION CONCENTRATED NEAR THE AXIS OF A DEEP-WATER WAVEGUIDE IN A RANGE-DEPENDENT MEDIUM

2006 ◽  
Vol 14 (02) ◽  
pp. 237-263 ◽  
Author(s):  
NATALIE S. GRIGORIEVA ◽  
GREGORY M. FRIDMAN
Keyword(s):  
Differential Equations ◽  
Helmholtz Equation ◽  
Deep Water ◽  
Deterministic Model ◽  
Geographic Location ◽  
Parabolic Cylinder ◽  
Left Hand ◽  
Residual Difference ◽  
Recurrent System ◽  
Limiting Case

This paper examines a two-dimensional deep-water waveguide in a range-dependent medium. Solutions of the homogeneous Helmholtz equation concentrated near the waveguide axis and which decrease exponentially outside a strip containing the axis are constructed. These solutions have the form of exponentials multiplied by parabolic cylinder functions whose arguments are an infinite series in powers of ω-1/2, where ω is a cyclic frequency. Coefficients of these series are found from a recurrent system of differential equations up to terms allowing to get a residual (difference between the left-hand side of the Helmholtz equation and zero) of order ω-1/2. Arbitrary constants of general solutions of arising differential equations are calculated going over to the limiting case of a range-independent medium. Numerical simulations are carried out for a deterministic model of a range-dependent ocean, where the range-dependence results for change in geographic location. Parameters of this medium model are matched with the AET experiment.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF THE HELMHOLTZ EQUATION CONCENTRATED NEAR THE AXIS OF A DEEP-WATER WAVEGUIDE IN A RANGE-INDEPENDENT MEDIUM

2004 ◽  
Vol 12 (01) ◽  
pp. 67-83 ◽  
Author(s):  
NATALIE S. GRIGORIEVA ◽  
GREGORY M. FRIDMAN
Keyword(s):  
Helmholtz Equation ◽  
Deep Water ◽  
Normal Modes ◽  
Principal Term ◽  
Parabolic Cylinder ◽  
Asymptotic Behavior Of Solutions ◽  
Narrow Strip ◽  
Left Hand ◽  
Residual Difference ◽  
Separable Case

To derive the integral representation for the axial wave describing the interference of near-axial waves in an arbitrary deep-water waveguide in a range-independent medium in long-range acoustic propagation in the ocean, it is necessary to construct the solutions of the homogeneous Helmholtz equation concentrated near the waveguide axis in a narrow strip having a width of order ω-1/2, where ω is a cyclic frequency. In a range-independent (separable) case the desired solutions coincide with the principal term of the uniform asymptotic expansion as ω→∞ of normal modes when a mode number is a value of order of unity. In this paper, the solutions of the homogeneous Helmholtz equation concentrated near the waveguide axis in a range-independent ocean and which decrease exponentially outside a strip containing the axis are constructed in the form admitting generalization to the case of a range-dependent medium. The solutions are represented as the product of exponentials and parabolic cylinder functions whose arguments are infinite series in powers of ω-1/2. Coefficients of these series are found from a recurrent system of partial differential equations up to terms allowing to get a residual (difference between the left-hand side of the Helmholtz equation and zero) of order ω-1/2. Numerical results are obtained for medium parameters corresponding to the Munk canonical sound-speed profile.

scholarly journals Efficient Approximate Techniques for Integrating Stochastic Differential Equations

2006 ◽  
Vol 134 (10) ◽  
pp. 3006-3014 ◽  
Author(s):  
James A. Hansen ◽  
Cecile Penland
Keyword(s):  
Time Series ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Deterministic Model ◽  
Probability Distributions ◽  
Approximate Solutions ◽  
Integration Scheme ◽  
Time Step ◽  
Integration Schemes ◽  
Runge Kutta

Abstract The delicate (and computationally expensive) nature of stochastic numerical modeling naturally leads one to look for efficient and/or convenient methods for integrating stochastic differential equations. Concomitantly, one may wish to sensibly add stochastic terms to an existing deterministic model without having to rewrite that model. In this note, two possibilities in the context of the fourth-order Runge–Kutta (RK4) integration scheme are examined. The first approach entails a hybrid of deterministic and stochastic integration schemes. In these examples, the hybrid RK4 generates time series with the correct climatological probability distributions. However, it is doubtful that the resulting time series are approximate solutions to the stochastic equations at every time step. The second approach uses the standard RK4 integration method modified by appropriately scaling stochastic terms. This is shown to be a special case of the general stochastic Runge–Kutta schemes considered by Ruemelin and has global convergence of order one. Thus, it gives excellent results for cases in which real noise with small but finite correlation time is approximated as white. This restriction on the type of problems to which the stochastic RK4 can be applied is strongly compensated by its computational efficiency.

A Mechanical Model of Helical Elements in Textured Yarns

1976 ◽  
Vol 46 (4) ◽  
pp. 278-283 ◽  
Author(s):  
M. Konopasek
Keyword(s):  
Differential Equations ◽  
Mechanical Model ◽  
Mechanical Characteristics ◽  
Functional Relationship ◽  
Three Dimensional ◽  
Infinite Length ◽  
Fiber Properties ◽  
Helical Angle ◽  
Limiting Case ◽  
Special Case

The helical model of the spontaneously collapsed filaments in twist-textured yarns is defined as reflecting the limiting case of a free-filament segment with infinite length (or number of coils) between two reversal points. The fundamental relationships linking fiber properties and parameters of the texturing process with geometrical and mechanical characteristics of the helices are derived directly from the differential equations of the three-dimensional elastica. Bicomponent and similar fibers are interpreted as a special case of twist-textured filaments with original (permanently set) helical angle equal to π/2; for this case an explicit functional relationship between contraction and stretching force is obtained.

Asymptotic behavior of a generalization of Bailey's simple epidemic

1971 ◽  
Vol 3 (02) ◽  
pp. 220-221
Author(s):  
George H. Weiss ◽  
Menachem Dishon
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Rate Constants ◽  
Deterministic Model ◽  
Stochastic Theory ◽  
Expected Number ◽  
Deterministic Theory ◽  
Mean Values ◽  
Image Position ◽  
Fixed Population

It has been shown that for many epidemic models, the stochastic theory leads to essentially the same results as the deterministic theory provided that one identifies mean values with the functions calculated from the deterministic differential equations (cf. [1]). If one considers a generalization of Bailey's simple epidemic for a fixed population of size N, represented schematically by where I refers to an infected, S refers to a susceptible, and α and β are appropriate rate constants, then it is evident that at time t = ∞, the expected number of infected individuals must be zero provided that β > 0. If x(t) denotes the number of infected at time t, then the deterministic model is summarized by

Derivation of Green-type, transitional, and uniform asymptotic expansions from differential equations II. Whittaker functions W k,m for large k , and for large | K 2 – m 2 |

1964 ◽  
Vol 281 (1384) ◽  
pp. 111-129 ◽  
Keyword(s):  
Differential Equations ◽  
Asymptotic Expansions ◽  
Mellin Transform ◽  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Whittaker Functions ◽  
Transform Method ◽  
Special Cases ◽  
Uniform Expansions ◽  
Uniform Asymptotic Expansions

The method for deriving Green-type asymptotic expansions from differential equations, introduced in I and illustrated therein by detailed calculations on modified Bessel functions, is applied to Whittaker functions W k,m , first for large k , and then for large |k 2 —m 2 |. Following the general theory of I, combination of this procedure with the Mellin transform method yields asymptotic expansions valid in transitional regions, and general uniform expansions. Weber parabolic cylinder and Poiseuille functions are examined as important special cases.

scholarly journals A Formal Time-domain Approach to Cold Magnetised Plasmas

1988 ◽  
Vol 41 (1) ◽  
pp. 55 ◽  
Author(s):  
Werner Weiglhofer
Keyword(s):  
Differential Equations ◽  
Time Domain ◽  
Scalar Potential ◽  
Coupled System ◽  
Dielectric Tensor ◽  
Tensor Operator ◽  
Transient Wave ◽  
The Time Domain ◽  
Limiting Case ◽  
Wave Phenomena

Representations of the electromagnetic and the average velocity field for a cold magnetised plasma are derived in terms of scalar potential functions. These Hertz potentials are solutions of a coupled system of integro-differential equations of second Qrder. Different from other approaches, the analysis is carried out in the time domain and is therefore especially suited for the investigation of transient wave phenomena. Furthermore, the dielectric tensor operator of the plasma is derived. Mter solving the system of integro-differential equations for a special limiting case, the applicability of the method presented is demonstrated and generalisations are discussed.

scholarly journals Genetic diversity of Microsporidia in the circulatory system of endemic amphipods from different locations and depths of ancient Lake Baikal

PeerJ ◽  
2018 ◽  
Vol 6 ◽  
pp. e5329 ◽  
Author(s):  
Mariya Dimova ◽  
Ekaterina Madyarova ◽  
Anton Gurkov ◽  
Polina Drozdova ◽  
Yulia Lubyaga ◽  
...  
Keyword(s):  
Genetic Diversity ◽  
Lake Baikal ◽  
Deep Water ◽  
Endemic Species ◽  
Sequence Data ◽  
Molecular Genetic ◽  
Circulatory System ◽  
Geographic Location ◽  
Small Subunit ◽  
Amphipod Species

Endemic amphipods (Amphipoda, Crustacea) of the most ancient and large freshwater Lake Baikal (Siberia, Russia) are a highly diverse group comprising >15% of all known species of continental amphipods. The extensive endemic biodiversity of Baikal amphipods provides the unique opportunity to study interactions and possible coevolution of this group and their parasites, such as Microsporidia. In this study, we investigated microsporidian diversity in the circulatory system of 22 endemic species of amphipods inhabiting littoral, sublittoral and deep-water zones in all three basins of Lake Baikal. Using molecular genetic techniques, we found microsporidian DNA in two littoral (Eulimnogammarus verrucosus,Eulimnogammarus cyaneus), two littoral/sublittoral (Pallasea cancellus,Eulimnogammarus marituji) and two sublittoral/deep-water (Acanthogammarus lappaceus longispinus,Acanthogammarus victorii maculosus) endemic species. Twenty sequences of the small subunit ribosomal (SSU) rDNA were obtained from the haemolymph of the six endemic amphipod species sampled from 0–60 m depths at the Southern Lake Baikal’s basin (only the Western shore) and at the Central Baikal. They form clusters with similarity toEnterocytospora,Cucumispora,Dictyocoela, and several unassigned Microsporidia sequences, respectively. Our sequence data show similarity to previously identified microsporidian DNA from inhabitants of both Lake Baikal and other water reservoirs. The results of our study suggest that the genetic diversity of Microsporidia in haemolymph of endemic amphipods from Lake Baikal does not correlate with host species, geographic location or depth factors but is homogeneously diverse.

Asymptotic behavior of a generalization of Bailey's simple epidemic

1971 ◽  
Vol 3 (2) ◽  
pp. 220-221 ◽  
Author(s):  
George H. Weiss ◽  
Menachem Dishon
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Rate Constants ◽  
Deterministic Model ◽  
Stochastic Theory ◽  
Expected Number ◽  
Deterministic Theory ◽  
Mean Values ◽  
Image Position ◽  
Fixed Population

It has been shown that for many epidemic models, the stochastic theory leads to essentially the same results as the deterministic theory provided that one identifies mean values with the functions calculated from the deterministic differential equations (cf. [1]). If one considers a generalization of Bailey's simple epidemic for a fixed population of size N, represented schematically by where I refers to an infected, S refers to a susceptible, and α and β are appropriate rate constants, then it is evident that at time t = ∞, the expected number of infected individuals must be zero provided that β > 0. If x(t) denotes the number of infected at time t, then the deterministic model is summarized by

Novel solutions of the Helmholtz equation and their application to diffraction

2007 ◽  
Vol 463 (2080) ◽  
pp. 1005-1027 ◽  
Author(s):  
Bair V Budaev ◽  
David B Bogy
Keyword(s):  
Differential Equations ◽  
Helmholtz Equation ◽  
Broad Class ◽  
Probabilistic Approach ◽  
Three Dimensional ◽  
Boundary Function ◽  
Original Method ◽  
Random Motion ◽  
Scattering Problems ◽  
Homogeneous Media

This paper presents a series of novel representations for the solutions of the Helmholtz equation in a broad class of wedge-like domains including those with curvilinear, non-flat faces. These representations are obtained by an original method which combines ray theory with the probabilistic approach to partial differential equations and uses a specific technique to deal with a need for analytical continuation of the specified boundary function. The main results are reminiscent of the standard Feynman–Kac formula but differ in that the averaging over solutions of stochastic differential equations is replaced by averaging over the trajectories of a new two-scale random motion introduced here. The paper focuses on the development of the solutions and for this reason it includes only a brief outline of numerous applications, consequences, extensions and variations of the method, which include, but are not limited to, problems of diffraction and scattering, problems in three-dimensional domains and problems of wave propagation in non-homogeneous media.

scholarly journals A reflection principle for solutions to the Helmholtz equation and an application to the inverse scattering problem

1977 ◽  
Vol 18 (2) ◽  
pp. 125-130 ◽  
Author(s):  
David Colton
Keyword(s):  
Helmholtz Equation ◽  
Analytic Continuation ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Reflection Principle ◽  
Dimensional Euclidean Space ◽  
Finite Radius ◽  
Spherical Boundary ◽  
Limiting Case

A classical result in potential theory is the Schwarz reflection principle for solutions of Laplace's equation which vanish on a portion of a spherical boundary. The question naturally arises whether or not such a property is also true for solutions of the Helmholtz equation. This has been answered in the affirmative by Diaz and Ludford ([4]; see also [10]) in the limiting case of the plane. It is the purpose of this paper to show that a reflection principle is also valid for spheres of finite radius. As an application of this result we shall study the problem of the analytic continuation of solutions to the Helmholtz equation defined in the exterior of a bounded domain in three-dimensional Euclidean space ℝ3 We shall show that through the use of the reflection principle derived in this paper, this problem can be reduced to the problem of the analytic continuation of an analytic function of two complex variables, which in turn can be performed through a variety of known methods (cf. [7]).

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