Time-Harmonic Waves Traveling Obliquely in a Periodically Laminated Medium

1971 ◽  
Vol 38 (2) ◽  
pp. 477-482 ◽  
Author(s):  
C. Sve
Keyword(s):  
Dispersion Relation ◽  
Fourth Order ◽  
Continuum Theory ◽  
Numerical Results ◽  
Harmonic Waves ◽  
Two Dimensional ◽  
Arbitrary Direction ◽  
Arbitrary Angle ◽  
Phase Velocities ◽  
Time Harmonic

The dispersion relation is presented for time-harmonic waves propagating in an arbitrary direction in a periodically laminated medium. The analysis is based on two-dimensional equations of elasticity. Limiting phase velocities are presented for infinite wavelength for any angle of propagation in the form of a fourth-order determinant that illustrates the influence of an arbitrary angle. For the cases when the propagation is along or across the layers, the determinant reduces to two determinants of second order that yield the limiting phase velocities directly. Numerical results are presented to indicate the dependence of dispersion upon the angle of propagation. Also, a comparison with an approximate continuum theory is included; agreement is satisfactory for those angles where the dispersion is the strongest.

Numerical Simulation of Time-Harmonic Waves in Inhomogeneous Media using Compact High Order Schemes

2011 ◽  
Vol 9 (3) ◽  
pp. 520-541 ◽  
Author(s):  
Steven Britt ◽  
Semyon Tsynkov ◽  
Eli Turkel
Keyword(s):  
Fourth Order ◽  
Finite Difference Methods ◽  
Accurate Method ◽  
Variable Coefficients ◽  
High Order ◽  
Harmonic Waves ◽  
Order Convergence ◽  
High Order Schemes ◽  
Time Harmonic ◽  
Difference Methods

AbstractIn many problems, one wishes to solve the Helmholtz equation with variable coefficients within the Laplacian-like term and use a high order accurate method (e.g., fourth order accurate) to alleviate the points-per-wavelength constraint by reducing the dispersion errors. The variation of coefficients in the equation may be due to an inhomogeneous medium and/or non-Cartesian coordinates. This renders existing fourth order finite difference methods inapplicable. We develop a new compact scheme that is provably fourth order accurate even for these problems. We present numerical results that corroborate the fourth order convergence rate for several model problems.

Guided Circumferential Waves in a Circular Annulus

1998 ◽  
Vol 65 (2) ◽  
pp. 424-430 ◽  
Author(s):  
Guoli Liu ◽  
Jianmin Qu
Keyword(s):  
Frequency Dispersion ◽  
Radius Of Curvature ◽  
Harmonic Waves ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Circular Annulus ◽  
Time Harmonic ◽  
Steady State Time ◽  
Wave Phenomena ◽  
Circumferential Waves

A two-dimensional circular annulus is considered in this paper as a waveguide. The guided steady-state time-harmonic waves propagating in the circumferential direction are studied. It is found that the guided circumferential waves are dispersive. The dispersion equation is derived analytically and numerical examples are presented for the frequency dispersion curves. The displacement profiles across the wall thickness of the annulus are also obtained for the first five propagating modes. In addition, the analogy between a flat plate and an annulus in the asymptotic limit of infinite radius of curvature is discussed to reveal some interesting wave phenomena intrinsic to curved waveguides.

Continuum Theory for a Laminated Medium

1968 ◽  
Vol 35 (3) ◽  
pp. 467-475 ◽  
Author(s):  
C.-T. Sun ◽  
J. D. Achenbach ◽  
George Herrmann
Keyword(s):  
Strain Energy ◽  
Equations Of Motion ◽  
Continuum Theory ◽  
Laminated Composite ◽  
Harmonic Waves ◽  
Approximate Theory ◽  
Lagrangian Multipliers ◽  
Phase Velocities ◽  
Wave Numbers ◽  
The Matrix

A system of displacement equations of motion is presented, pertaining to a continuum theory to describe the dynamic behavior of a laminated composite. In deriving the equations, the displacements of the reinforcing layers and the matrix layers are expressed as two-term expansions about the mid-planes of the layers. Dynamic interaction of the layers is included through continuity relations at the interfaces. By means of a smoothing operation, representative kinetic and strain energy densities for the laminated medium are obtained. Subsequent application of Hamilton’s principle, where the continuity relations are included through the use of Lagrangian multipliers, yields the displacement equations of motion. The distinctive trails of the system of equations are uncovered by considering the propagation of plane harmonic waves. Dispersion curves for harmonic waves propagating parallel to and normal to the layering are presented, and compared with exact curves. The limiting phase velocities at vanishing wave numbers agree with the exact, limits. The lowest antisymmetric mode for waves propagating in the direction of the layering shows the strongest dispersion, which is very well described by the approximate theory over a substantial range of wave numbers.

Time-Harmonic Waves Propagating Normal to the Layers of Multilayered Periodic Media

1974 ◽  
Vol 41 (1) ◽  
pp. 92-96 ◽  
Author(s):  
Adnan H. Nayfeh
Keyword(s):  
Elastic Modulus ◽  
Dispersion Relation ◽  
Mass Density ◽  
Harmonic Waves ◽  
Periodic Media ◽  
Incident Wavelength ◽  
Time Harmonic ◽  
Periodic Composite ◽  
Special Cases

The dispersion relation is derived for time-harmonic waves propagating normal to the layers of a multilayered periodic composite. The known relations for the homogeneous and the bilaminated media are deduced as special cases. Mixture mass density and elastic modulus are defined in the limit as the ratio of the incident wavelength to the microdimension of the composite approaches infinity.

Theory of Laminated Plates

1971 ◽  
Vol 38 (1) ◽  
pp. 231-238 ◽  
Author(s):  
C. T. Sun
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Continuum Theory ◽  
Laminated Plates ◽  
Harmonic Waves ◽  
Two Dimensional ◽  
Laminated Plate ◽  
Governing Equations ◽  
Dimensional Theory ◽  
Rotatory Inertia

A two-dimensional theory for laminated plates is deduced from the three-dimensional continuum theory for a laminated medium. Plate-stress equations of motion, plate-stress-strain relations, boundary conditions, and plate-displacement equations of motion are presented. The governing equations are employed to study the propagation of harmonic waves in a laminated plate. Dispersion curves are presented and compared with those obtained according to the three-dimensional continuum theory and the exact analysis. An approximate solution for flexural motions obtained by neglecting the gross and local rotatory inertia terms is also discussed.

scholarly journals Onset of Convection in Two-Dimensional Porous Cavities with Open and Conducting Boundaries

2021 ◽  
Vol 136 (3) ◽  
pp. 791-812
Author(s):  
Peder A. Tyvand ◽  
Jonas Kristiansen Nøland
Keyword(s):  
Critical Rayleigh Number ◽  
Numerical Results ◽  
Temperature Perturbation ◽  
Two Dimensional ◽  
Onset Of Convection ◽  
Order Problem ◽  
Fourth Order Problem ◽  
Thermally Conducting ◽  
Perturbation Pressure ◽  
Special Case

AbstractThe onset of thermal convection in two-dimensional porous cavities heated from below is studied theoretically. An open (constant-pressure) boundary is assumed, with zero perturbation temperature (thermally conducting). The resulting eigenvalue problem is a full fourth-order problem without degeneracies. Numerical results are presented for rectangular and elliptical cavities, with the circle as a special case. The analytical solution for an upright rectangle confirms the numerical results. Streamlines penetrating the open cavities are plotted, together with the isotherms for the associated closed thermal cells. Isobars forming pressure cells are depicted for the perturbation pressure. The critical Rayleigh number is calculated as a function of geometric parameters, including the tilt angle of the rectangle and ellipse. An improved physical scaling of the Darcy–Bénard problem is suggested. Its significance is indicated by the ratio of maximal vertical velocity to maximal temperature perturbation.

Numerical Analysis of Two-Phase Pipe Flow of Liquid Helium Using Multi-Fluid Model

2001 ◽  
Vol 123 (4) ◽  
pp. 811-818 ◽  
Author(s):  
Jun Ishimoto ◽  
Mamoru Oike ◽  
Kenjiro Kamijo
Keyword(s):  
Phase Transition ◽  
Gas Phase ◽  
Liquid Helium ◽  
Two Phase Flow ◽  
Fluid Model ◽  
Numerical Results ◽  
Phase Flow ◽  
Two Dimensional ◽  
Two Phase ◽  
Normal Fluid

The two-dimensional characteristics of the vapor-liquid two-phase flow of liquid helium in a pipe are numerically investigated to realize the further development and high performance of new cryogenic engineering applications. First, the governing equations of the two-phase flow of liquid helium based on the unsteady thermal nonequilibrium multi-fluid model are presented and several flow characteristics are numerically calculated, taking into account the effect of superfluidity. Based on the numerical results, the two-dimensional structure of the two-phase flow of liquid helium is shown in detail, and it is also found that the phase transition of the normal fluid to the superfluid and the generation of superfluid counterflow against normal fluid flow are conspicuous in the large gas phase volume fraction region where the liquid to gas phase change actively occurs. Furthermore, it is clarified that the mechanism of the He I to He II phase transition caused by the temperature decrease is due to the deprivation of latent heat for vaporization from the liquid phase. According to these theoretical results, the fundamental characteristics of the cryogenic two-phase flow are predicted. The numerical results obtained should contribute to the realization of advanced cryogenic industrial applications.

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