Numerical Simulation of Cylindrical Solitary Waves in Periodic Media

Luna, Manuel Quezada de; Ketcheson, David I.

doi:10.1007/s10915-013-9747-3

Cited by 8 publications

(17 citation statements)

References 15 publications

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“…Since the boundary conditions are fulfilled by the leading order homogenized system (11), the third homogenized correction vanishes; i.e.,ū…”

Section: O(δ 3 ) Homogenized Correctionmentioning

confidence: 99%

Two-Dimensional Wave Propagation in Layered Periodic Media

Luna¹,

Ketcheson²

2014

SIAM J. Appl. Math.

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We study two-dimensional wave propagation in materials whose properties vary periodically in one direction only. High order homogenization is carried out to derive a dispersive effective medium approximation. One-dimensional materials with constant impedance exhibit no effective dispersion. We show that a new kind of effective dispersion may arise in two dimensions, even in materials with constant impedance. This dispersion is a macroscopic effect of microscopic diffraction caused by spatial variation in the sound speed. We analyze this dispersive effect by using highorder homogenization to derive an anisotropic, dispersive effective medium. We generalize to two dimensions a homogenization approach that has been used previously for one-dimensional problems. Pseudospectral solutions of the effective medium equations agree to high accuracy with finite volume direct numerical simulations of the variable-coefficient equations.

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“…Since the boundary conditions are fulfilled by the leading order homogenized system (11), the third homogenized correction vanishes; i.e.,ū…”

Section: O(δ 3 ) Homogenized Correctionmentioning

confidence: 99%

Two-Dimensional Wave Propagation in Layered Periodic Media

Luna¹,

Ketcheson²

2014

SIAM J. Appl. Math.

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“…Propagation of acoustic and elastic waves in nonlinear phononic crystals has attracted great attention [1][2][3][4][5], because understanding of these dynamic behaviors not only offers the possibility of realizing nonlinear phenomena such as solitons [6][7][8][9] but also provides a powerful tool to manipulate the waves via nonlinearity [10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Solitary waves in a granular chain of elastic spheres: Multiple solitary solutions and their stabilities

2019

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A granular chain of elastic spheres via Hertzian contact incorporates a classical nonlinear force model describing dynamical elastic solitary wave propagation. In this paper, the multiple solitary waves and their dynamic behaviors and stability in such a system are considered. An approximate KdV equation with the standard form is derived under the long-wavelength approximation and small deformation. The closed-form analytical single-and multiple-soliton solutions are obtained. The construction of the multiple-soliton solutions is analyzed by using the functional analysis. It is found that the multiple-soliton solution can be excited by the single-soliton solutions. This result is confirmed by the numerical analysis. Based on the soliton solutions of the KdV equation, the analytic solutions of multiple dark solitary waves are obtained from the original dynamic equation of the granular chain in the long-wavelength approximation. The stability of the single and multiple dark solitary wave solutions are numerically analyzed by using both split-step Fourier transform method and Runge-Kutta method. The results show that the single dark solitary wave solution is stable, and the multiple dark solitary waves are unstable.

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“…which may be viewed as a multi-dimensional analog of the p-system. We use the notation of elasticity, for consistency with related work [10,12]; thus is the strain, ρ is the density, and σ is the stress. If the stress-strain function is linear, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…By symmetry, the problem can be solved by considering a single period of the medium and periodic boundary conditions in y. We compute solutions to (1) using the finite volume solver PyClaw [8,9] with the Riemann solvers described in [12]. We consider the solution after the perturbation has travelled a distance of more than 300 material periods.…”

Section: Introductionmentioning

confidence: 99%

Diffractons: Solitary Waves Created by Diffraction in Periodic Media

Ketcheson¹,

Luna²

2015

Multiscale Model. Simul.

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A new class of solitary waves arises in the solution of nonlinear wave equations with constant impedance and no dispersive terms. They depend on a balance between nonlinearity and a dispersion-like effect due to spatial variation in the sound speed of the medium. A high-order homogenized model confirms this effective dispersive behavior and its solutions agree well with those obtained by direct simulation of the variable-coefficient system. These waves are observed to be long-time stable, globally attracting solutions that arise in general as solutions to nonlinear wave problems with periodically-varying sound speed. They share some properties with known classes of solitary waves, but possess important differences as well.

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Numerical Simulation of Cylindrical Solitary Waves in Periodic Media

Cited by 8 publications

References 15 publications

Two-Dimensional Wave Propagation in Layered Periodic Media

Two-Dimensional Wave Propagation in Layered Periodic Media

Solitary waves in a granular chain of elastic spheres: Multiple solitary solutions and their stabilities

Diffractons: Solitary Waves Created by Diffraction in Periodic Media

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