scholarly journals Coupled mode theory of scattering by a cylindrically symmetric seamount

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150465 ◽  
Author(s):  
Ronald F. Pannatoni
Keyword(s):  
Three Dimensional ◽  
Numerical Procedure ◽  
Infinite Dimensional ◽  
Coupled Mode ◽  
Cylindrically Symmetric ◽  
Finite Dimensional ◽  
Time Harmonic ◽  
Acoustic Waveguides ◽  
Algebraic Formula ◽  
Stratified Ocean

The coupled-mode equations of Shevchenko are extended to three-dimensional irregular acoustic waveguides. These equations are solved for a model of a cylindrically symmetric seamount and a time-harmonic point source in a horizontally stratified ocean. An algebraic formula for the scattered field beyond the seamount is obtained from this solution. The formula is characterized by a set of infinite-dimensional matrices that are independent of the source. A stable numerical procedure is developed to compute finite-dimensional approximations to these matrices for use in truncated versions of the formula.

scholarly journals THE QUANTUM H3 INTEGRABLE SYSTEM

2010 ◽  
Vol 25 (30) ◽  
pp. 5567-5594 ◽  
Author(s):  
MARCOS A. G. GARCÍA ◽  
ALEXANDER V. TURBINER
Keyword(s):  
Integrable System ◽  
Differential Operators ◽  
Three Dimensional ◽  
Integrable Model ◽  
Dimensional System ◽  
Property A ◽  
Algebraic Form ◽  
Infinite Dimensional ◽  
Polynomial Coefficients ◽  
Finite Dimensional

The quantum H3 integrable system is a three-dimensional system with rational potential related to the noncrystallographic root system H3. It is shown that the gauge-rotated H3 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H3, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector [Formula: see text]. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H3 Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of H3-invariants "polynomially"-isospectral to the quantum H3 model is defined.

A Coupled-Mode Method for Acoustic Propagation and Scattering in Inhomogeneous Ocean Waveguides

2014 ◽  
Author(s):  
K. A. Belibassakis ◽  
G. A. Athanassoulis
Keyword(s):  
Bottom Topography ◽  
Three Dimensional ◽  
Acoustic Propagation ◽  
Local Mode ◽  
Scattering Problems ◽  
Coupled Mode ◽  
Cylindrically Symmetric ◽  
Present Solution ◽  
Mode Method ◽  
Additional Mode

We consider the problem of acoustic propagation and scattering in inhomogeneous waveguide governed by the Helmholtz equation. We focus on an ideal, cylindrically symmetric ocean waveguide, limited above by an acoustically soft boundary modelling the free surface, and below by a hard boundary modelling the impenetrable seabed with general bottom topography. The wave field is excited by a monochromatic point source, and thus, the present solution is equivalent to the construction of the Green’s function in the inhomogeneous domain. An improved coupled-mode method is developed, based on an enhanced local-mode series for the representation of the acoustic field, which includes an additional mode accounting for the effects of the bottom slope and curvature. The additional mode provides an implicit summation of the slowly convergent part of the series, rendering the remaining part to converge much faster, pemitting truncation of the modal expansions keeping only a few evanescent terms. Using the enhanced representation, in conjunction with an appropriate variational principle, a system of coupled-mode equations on the horizontal plane is derived for the determination of the complex modal-amplitude functions. Numerical results are presented including comparisons with analytical solutions illustrating the role and significance of the additional mode and the efficiency of the present coupled-mode tmodel, which can be naturally extended to treat propagation and scattering problems in three-dimensional, multi-layered ocean acoustic waveguides.

A Numerical Procedure for Vehicle-Induced Internal Waves

1988 ◽  
Vol 110 (4) ◽  
pp. 380-386
Author(s):  
I. Suh ◽  
R. I. Hires
Keyword(s):  
Steady State ◽  
Internal Waves ◽  
Axial Velocity ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Velocity Variation ◽  
Cartesian Coordinate System ◽  
Direct Solution ◽  
Cartesian Coordinate ◽  
Stratified Ocean

An axially marching numerical method is developed for the simulation of the internal waves produced by the translation of a submersed vehicle in a density-stratified ocean. The method provides for the direct solution of the primitive variables [v, p, ρ] for the nonlinear and steady-state three-dimensional Euler’s equation with a nonconstant density term in the vehicle-fixed cartesian coordinate system. By utilizing a known potential flow around the vehicle for an estimate of the axial velocity gradient, the present parabolic algorithm allows local upstream disturbances and an axial velocity variation.

COUPLED MODE AND FINITE ELEMENT APPROXIMATIONS OF UNDERWATER SOUND PROPAGATION PROBLEMS IN GENERAL STRATIFIED ENVIRONMENTS

2008 ◽  
Vol 16 (01) ◽  
pp. 83-116 ◽  
Author(s):  
G. A. ATHANASSOULIS ◽  
K. A. BELIBASSAKIS ◽  
D. A. MITSOUDIS ◽  
N. A. KAMPANIS ◽  
V. A. DOUGALIS
Keyword(s):  
Finite Element ◽  
Sound Propagation ◽  
Test Problems ◽  
Underwater Sound ◽  
Finite Element Approximations ◽  
Coupled Mode ◽  
Cylindrically Symmetric ◽  
Mode Method ◽  
Acoustic Waveguides ◽  
Good Agreement

We compare the results of a coupled mode method with those of a finite element method and also of COUPLE on two test problems of sound propagation and scattering in cylindrically symmetric, underwater, multilayered acoustic waveguides with range-dependent interface topographies. We observe, in general, very good agreement between the results of the three codes. In some cases in which the frequency of the harmonic point source is such that an eigenvalue of the local vertical problem remains small in magnitude and changes sign several times in the vicinity of the interface nonhomogeneity, the discrepancies between the results of the three codes increase, but remain small in absolute terms.

scholarly journals The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160585 ◽  
Author(s):  
Divya Venkataraman ◽  
Samriddhi Sankar Ray
Keyword(s):  
Energy Spectrum ◽  
Numerical Simulations ◽  
Euler Equations ◽  
Burgers Equation ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Infinite Dimensional ◽  
Localized Structures ◽  
Finite Dimensional ◽  
Incompressible Euler

Solutions to finite-dimensional (all spatial Fourier modes set to zero beyond a finite wavenumber K G ), inviscid equations of hydrodynamics at long times are known to be at variance with those obtained for the original infinite dimensional partial differential equations or their viscous counterparts. Surprisingly, the solutions to such Galerkin-truncated equations develop sharp localized structures, called tygers (Ray et al. 2011 Phys. Rev. E 84 , 016301 ( doi:10.1103/PhysRevE.84.016301 )), which eventually lead to completely thermalized states associated with an equipartition energy spectrum. We now obtain, by using the analytically tractable Burgers equation, precise estimates, theoretically and via direct numerical simulations, of the time τ c at which thermalization is triggered and show that τ c ∼ K G ξ , with ξ = − 4 9 . Our results have several implications, including for the analyticity strip method, to numerically obtain evidence for or against blow-ups of the three-dimensional incompressible Euler equations.

scholarly journals An FDA-Based Approach for Clustering Elicited Expert Knowledge

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 184-204
Author(s):  
Carlos Barrera-Causil ◽  
Juan Carlos Correa ◽  
Andrew Zamecnik ◽  
Francisco Torres-Avilés ◽  
Fernando Marmolejo-Ramos
Keyword(s):  
Functional Data ◽  
Expert Knowledge ◽  
Probability Distributions ◽  
Knowledge Elicitation ◽  
Parallel Analysis ◽  
Data Sets ◽  
Dimensional Problem ◽  
Prior Distributions ◽  
Infinite Dimensional ◽  
Finite Dimensional

Expert knowledge elicitation (EKE) aims at obtaining individual representations of experts’ beliefs and render them in the form of probability distributions or functions. In many cases the elicited distributions differ and the challenge in Bayesian inference is then to find ways to reconcile discrepant elicited prior distributions. This paper proposes the parallel analysis of clusters of prior distributions through a hierarchical method for clustering distributions and that can be readily extended to functional data. The proposed method consists of (i) transforming the infinite-dimensional problem into a finite-dimensional one, (ii) using the Hellinger distance to compute the distances between curves and thus (iii) obtaining a hierarchical clustering structure. In a simulation study the proposed method was compared to k-means and agglomerative nesting algorithms and the results showed that the proposed method outperformed those algorithms. Finally, the proposed method is illustrated through an EKE experiment and other functional data sets.

scholarly journals On the forces that cable webs under tension can support and how to design cable webs to channel stresses

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180781 ◽  
Author(s):  
Guy Bouchitté ◽  
Ornella Mattei ◽  
Graeme W. Milton ◽  
Pierre Seppecher
Keyword(s):  
Linear Programming ◽  
Convex Hull ◽  
Structural Engineering ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Dimensional ◽  
Necessary And Sufficient

In many applications of structural engineering, the following question arises: given a set of forces f 1 ,  f 2 , …,  f N applied at prescribed points x 1 ,  x 2 , …,  x N , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x 1 ,  x 2 , …,  x N in the two- and three-dimensional cases. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two dimensions, we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f 1 ,  f 2 , …,  f N applied at points strictly within the convex hull of x 1 ,  x 2 , …,  x N . In three dimensions, we show that, by slightly perturbing f 1 ,  f 2 , …,  f N , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for channelling stress in desired ways.

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