Bulletin of the American Physical Society
74th Annual Meeting of the Southeastern Section
Volume 52, Number 13
Thursday–Saturday, November 8–10, 2007; Nashville, Tennessee
Session DB: Non-Linear Wave and Continuum Mechanics Phenomena |
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Chair: Ronald Mickens, Clark Atlanta University Room: Scarritt-Bennett Center Laskey C |
Thursday, November 8, 2007 1:30PM - 2:00PM |
DB.00001: On the Rational Continuum Mechanics of the Unified Field Invited Speaker: In recent author's works the argument has been made that a viscoelastic absolute medium (called the \emph{metacontinuum}) exists and it was shown that Maxwell-Hertz equations are a straightforward corollary from the governing equations of the \emph{metacontinuum}. Here we assume that the \emph{metacontinuum} is actually a thin 3D hypershell in the 4D space. The ``master'' equation for the deflection $\zeta$ of very thin but very stiff shells is a nonlinear equation of Boussinesq type. As mentioned by Schr\"{o}dinger himself the wave equation written for the real or imaginary part of the wave function is a linearized shell/plate equation. Thus the wave function in our model has a clear non-probabilistic interpretation as the \emph{actual} amplitude of the flexural deformation. A dispersive nonlinear equation admits solitary wave solutions (solitons) that behave as particles upon collisions (called quasi-particles or QPs). We stipulate here that the material particles are our perception (\emph{schaumkommen} in Schr\"{o}dinger's own words) of the QPs of the ``master'' equation. We show the passage from the continuous to the discrete Lagrangian of the centers of QPs and introduce the concept of (pseudo)mass. The membrane tension results into an attractive (gravitational?) force acting between the QPs. [Preview Abstract] |
Thursday, November 8, 2007 2:00PM - 2:30PM |
DB.00002: Nonlinear Phenomena in Acoustics: Traveling Waves, Bifurcations, and Singular Surfaces Invited Speaker: Traveling wave solutions (TWS) are explored in the context of nonlinear acoustics. Exact solutions are given, including one involving the recently introduced Lambert $W$-function, along with asymptotic and stability results. Poroacoustic propagation under Darcy's and Forchheimer's laws is examined, as well as acoustic phenomena in thermoviscous fluids. Additionally, a connection between discontinuity formation in the TWS and the associated singular surface, which is known as an acceleration wave, is pointed out. Lastly, if time permits, applications to nonlinear kinematic wave phenomena (e.g., second-sound and traffic flow) are briefly noted. [Preview Abstract] |
Thursday, November 8, 2007 2:30PM - 3:00PM |
DB.00003: A Hyperbolic Two-Step Model Based Finite Difference Scheme for Studying Thermal Deformation in a Double-Layered Thin Film Exposed to Ultrashort-Pulsed Lasers Invited Speaker: Hyperbolic two-step micro heat transport equations have attracted attention in thermal analysis of thin metal films exposed to ultrashort-pulsed lasers. In this talk, we present a finite difference scheme for studying thermal deformation in a 2D double-layered micro thin film exposed to ultrashort-pulsed lasers. This scheme is obtained based on the hyperbolic two-step model with temperature-dependent thermal properties. The method is illustrated by investigating the heat transfer in a gold layer padding on a chromium layer. Result shows that there are not non-physical oscillations in the solution. [Preview Abstract] |
Thursday, November 8, 2007 3:00PM - 3:30PM |
DB.00004: Nonstandard Finite Difference (NSTD) Schemes for Wave Equations: A Review Invited Speaker: Many phenomena in the natural and engineering sciences can be understood in terms of representations by partial differential equations. Particular examples of such equations include the linear and nonlinear unidirectional wave equations, Burger's equation, the Fisher equation, and the full wave equation. These PDE's can also include damping and reaction terms, along with nonlinear advection. Since few of these PDE's have known general solutions, numerical methods are the only practical way to obtain accurate solutions for given sets of initial or boundary conditions. A widely used procedure for calculating numerical solutions is finite differences. We demonstrate that the NSFD methodology provides a concise and dynamic consistent procedure for constructing finite difference schemes for wave equations. Our major focus is on PDE's for which a positivity condition holds. [Preview Abstract] |
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