scholarly journals FAST SOLITONS ON STAR GRAPHS

2011 ◽  
Vol 23 (04) ◽  
pp. 409-451 ◽  
Author(s):  
RICCARDO ADAMI ◽  
CLAUDIO CACCIAPUOTI ◽  
DOMENICO FINCO ◽  
DIEGO NOJA
Keyword(s):  
Solitary Wave ◽  
Boundary Conditions ◽  
Linear Problem ◽  
Cubic Nonlinearity ◽  
Point Interactions ◽  
Reflection And Transmission ◽  
Seminal Paper ◽  
Star Graphs ◽  
Transmission Coefficients ◽  
Energy Domain

We define the Schrödinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoff boundary condition and the so-called δ and δ′ boundary conditions. Moreover, in the same setting, we study the collision of a fast solitary wave with the vertex and we show that it splits in reflected and transmitted components. The outgoing waves preserve a soliton character over a time which depends on the logarithm of the velocity of the ingoing solitary wave. Over the same timescale, the reflection and transmission coefficients of the outgoing waves coincide with the corresponding coefficients of the linear problem. In the analysis of the problem, we follow ideas borrowed from the seminal paper [17] about scattering of fast solitons by a delta interaction on the line, by Holmer, Marzuola and Zworski. The present paper represents an extension of their work to the case of graphs and, as a byproduct, it shows how to extend the analysis of soliton scattering by other point interactions on the line, interpreted as a degenerate graph.

Wave Propagation Analysis of an Axially Strained, Rotating Timoshenko Shaft: Part II

1997 ◽  
Author(s):  
Bongsu Kang ◽  
Chin An Tan
Keyword(s):  
Boundary Conditions ◽  
Wave Reflection ◽  
Axial Strain ◽  
Free Vibration Analysis ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Propagation Analysis ◽  
Geometric Discontinuities ◽  
Bernoulli Models ◽  
Incident Waves

Abstract In this paper, the wave reflection and transmission characteristics of an axially strained, rotating Timoshenko shaft under general support and boundary conditions, and with geometric discontinuities are examined. As a continuation to Part I of this paper (Kang and Tan, 1997), the wave reflection and transmission at point supports with finite translational and rotational constraints are further discussed. The reflection and transmission matrices for incident waves upon general supports and geometric discontinuities are derived. These matrices are combined, with the aid of the transfer matrix method, to provide a concise and systematic approach for the free vibration analysis of multi-span rotating shafts with general boundary conditions. Results on the wave reflection and transmission coefficients are presented for both the Timoshenko and the Euler-Bernoulli models to investigate the effects of the axial strain, shaft rotation speed, shear and rotary inertia.

Boundary conditions for a fluid‐saturated porous solid

Geophysics ◽  
1987 ◽  
Vol 52 (2) ◽  
pp. 174-178 ◽  
Author(s):  
Oscar M. Lovera
Keyword(s):  
Boundary Conditions ◽  
Total Energy ◽  
Elastic Medium ◽  
Contact Surface ◽  
Normal Component ◽  
Porous Solid ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Contact Surfaces ◽  
Transmission And Reflection

Using Biot’s theory, a set of boundary conditions is presented for wave transmission and reflection at the contact surface between an elastic medium (fluid or solid) and a fluid‐saturated porous solid (Biot medium). The analysis shows the continuity of the normal component of the density energy flux vectors across the contact surfaces, so that total energy is preserved. Energy reflection and transmission coefficients are computed for each kind of Biot wave.

The Wave Propagation through the Imperfect Interface between Two Micropolar Solids

2013 ◽  
Vol 803 ◽  
pp. 419-422 ◽  
Author(s):  
Chong Fu ◽  
Pei Jun Wei
Keyword(s):  
Wave Propagation ◽  
Boundary Conditions ◽  
Algebraic Equation ◽  
Linear Algebraic Equation ◽  
Imperfect Interface ◽  
Transmission Problem ◽  
Transverse Displacement ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Rotational Waves

in this paper, the reflection and transmission problem of the coupled transverse displacement and transverse rotational waves at the imperfect interface between two different micropolar solids are studied. First, the boundary conditions between two micropolar solids with imperfect interface are used to derive the linear algebraic equation sets. Then, the linear algebraic equation sets are solved numerically and the results are shown graphically. Finally, the influence of the interface parameter reflecting the imperfect degree of interface on the reflection and the transmission coefficients are discussed based on the numerical results.Keyword: reflection and transmission, imperfect interface, micropolar solid, coupled waves.

Reflection and Refraction of Waves at a Planar Composite Interface

2011 ◽  
Author(s):  
Dale Chimenti ◽  
Stanislav Rokhlin ◽  
Peter Nagy
Keyword(s):  
Wave Propagation ◽  
Composite Materials ◽  
Boundary Conditions ◽  
Practical Importance ◽  
Ultrasonic Waves ◽  
Plane Interface ◽  
Contact Method ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Composite Interface

Nondestructive ultrasonic testing of composite materials is affected by several special features of wave propagation that arise from the strong anisotropy and inhomogeneity of these materials. The resulting complexity requires re-examination of old testing methodologies and development of new ones. One of the most fundamental phenomena in ultrasonic nondestructive evaluation is the reflection–refraction of ultrasonic waves at a plane interface. Even the simplest test procedure requires understanding of mode conversion and knowledge of elastic wave reflection and transmission coefficients and refraction angles. Reflection–refraction phenomena, while straightforward and well documented for isotropic materials, are much more complicated for anisotropic materials. When analyzing the oblique incidence inspection method for composite materials, one first has to address the problem of wave propagation through the interface between the coupling medium and the composite material. For example, there is an inherent fluid/composite interface in the immersion technique and a perspex/composite interface in the contact method. In the latter case, assuming that a thin fluid layer is applied to facilitate coupling through the interface, slip rather than welded boundary conditions prevail. Another example of great practical importance is the case of multidirectional fiber plies in a composite laminate, when the reflection and transmission of ultrasonic waves from one ply to another with a different orientation must be considered. Before discussing the general problem of wave refraction in anisotropic composite materials, let us review the simple isotropic case. Consider a plane interface between two isotropic elastic media in “welded” (perfectly bonded) contact, implying continuity of tractions and displacements across the interface, although the boundary conditions are not important at this point. Figure 4.1 shows a schematic diagram of a plane wave with wavenumber ki incident on the interface at angle θi. The parallel lines with spacing equal to the incident wavelength λi correspond to equal-phase planes orthogonal to the incident plane. By definition, the wavenumber ki = 2π/λi is the magnitude of the wave vector ki. The incident wave is converted at the interface into reflected and transmitted waves. The refraction angle of the transmitted wave is θr and its wavenumber is kr.

Extension of forward modeling phase-screen code in isotropic and anisotropic media up to critical angle

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM107-SM114 ◽  
Author(s):  
James C. White ◽  
Richard W. Hobbs
Keyword(s):  
Critical Angle ◽  
Model Space ◽  
Anisotropic Media ◽  
Forward Modeling ◽  
Phase Screen ◽  
Computationally Efficient ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Phase Corrections ◽  
Converted Phases

The computationally efficient phase-screen forward modeling technique is extended to allow investigation of nonnormal raypaths. The code is developed to accommodate all diffracted and converted phases up to critical angle, building on a geometric construction method. The new approach relies upon prescanning the model space to assess the complexity of each screen. The propagating wavefields are then divided as a function of horizontal wavenumber, and each subset is transformed to the spatial domain separately, carrying with it angular information. This allows both locally accurate 3D phase corrections and Zoeppritz reflection and transmission coefficients to be applied. The phase-screen code is further developed to handle simple anisotropic media. During phase-screen modeling, propagation is undertaken in the wavenumber domain where exact expressions for anisotropic phase velocities are available. Traveltimes and amplitude effects from a range of anisotropic shales are computed and compared with previous published results.

Reflection and Transmission Coefficients of Plane Waves in Magnetoelectroelastic Layered Structures

2008 ◽  
Vol 130 (3) ◽  
Author(s):  
J. Y. Chen ◽  
H. L. Chen ◽  
E. Pan
Keyword(s):  
Piezoelectric Materials ◽  
Plane Waves ◽  
Layered Structures ◽  
Organic Glass ◽  
Reflection And Transmission Coefficients ◽  
Material Structure ◽  
Layered System ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Novel Material

Reflection and transmission coefficients of plane waves with oblique incidence to a multilayered system of piezomagnetic and/or piezoelectric materials are investigated in this paper. The general Christoffel equation is derived from the coupled constitutive and balance equations, which is further employed to solve the elastic displacements and electric and magnetic potentials. Based on these solutions, the reflection and transmission coefficients in the corresponding layered structures are subsequently obtained by virtue of the propagator matrix method. Two layered examples are selected to verify and illustrate our solutions. One is the purely elastic layered system composed of aluminum and organic glass materials. The other layered system is composed of the novel magnetoelectroelastic material and the organic glass. Numerical results are presented to demonstrate the variation of the reflection and transmission coefficients with different incident angles, frequencies, and boundary conditions, which could be useful to nondestructive evaluation of this novel material structure based on wave propagations.

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