Closure to “Discussion of ‘Characteristic Forms of Differential Equations for Wave Propagation in Nonlinear Media’” (1982, ASME J. Appl. Mech., 49, p. 929)

1982 ◽  
Vol 49 (4) ◽  
pp. 930-930
Author(s):  
T. C. T. Ting
Keyword(s):  
Wave Propagation ◽  
Differential Equations ◽  
Nonlinear Media

Characteristic Forms of Differential Equations for Wave Propagation in Nonlinear Media

1981 ◽  
Vol 48 (4) ◽  
pp. 743-748 ◽  
Author(s):  
T. C. T. Ting
Keyword(s):  
Wave Propagation ◽  
Differential Equations ◽  
Constitutive Equations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Compressible Fluids ◽  
Elastic Solids ◽  
Nonlinear Media ◽  
Nonlinear Materials ◽  
One Dimensional

Characteristic forms of differential equations for wave propagation in nonlinear media are derived directly from equations of motion and equations which combine the constitutive equations and the equations of continuity. Both Lagrangian coordinates and Eulerian coordinates are considered. The constitutive equations considered here apply to a large class of nonlinear materials. The characteristic forms derived here clearly show which components of the stress and velocity are involved in the differentiation along the bicharacteristics. Moreover, the reduction to one-dimensional cases from three-dimensional problems is obvious for the characteristic forms obtained here. Examples are given and compared with the known solution in the literature for wave propagation in linear isotropic elastic solids and isentropic compressible fluids.

scholarly journals Analytical mathematical schemes: Circular rod grounded via transverse Poisson’s effect and extensive wave propagation on the surface of water

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 545-554
Author(s):  
Asghar Ali ◽  
Aly R. Seadawy ◽  
Dumitru Baleanu
Keyword(s):  
Wave Propagation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Longitudinal Wave ◽  
Symbolic Computation ◽  
Boussinesq Equation ◽  
Nonlinear Partial Differential Equations ◽  
Wave Model ◽  
Simple Equation ◽  
Partial Differential

AbstractThis article scrutinizes the efficacy of analytical mathematical schemes, improved simple equation and exp(-\text{Ψ}(\xi ))-expansion techniques for solving the well-known nonlinear partial differential equations. A longitudinal wave model is used for the description of the dispersion in the circular rod grounded via transverse Poisson’s effect; similarly, the Boussinesq equation is used for extensive wave propagation on the surface of water. Many other such types of equations are also solved with these techniques. Hence, our methods appear easier and faster via symbolic computation.

A Simple Three-States Cellular Automaton for Modelling Excitable Media

1998 ◽  
Vol 12 (05) ◽  
pp. 601-607 ◽  
Author(s):  
M. Andrecut
Keyword(s):  
Wave Propagation ◽  
Partial Differential Equations ◽  
Cellular Automata ◽  
Differential Equations ◽  
Cellular Automaton ◽  
Cellular Automaton Model ◽  
Excitable Media ◽  
Self Organization ◽  
Partial Differential ◽  
Bz Reaction

Wave propagation in excitable media provides an important example of spatiotemporal self-organization. The Belousov–Zhabotinsky (BZ) reaction and the impulse propagation along nerve axons are two well-known examples of this phenomenon. Excitable media have been modelled by continuous partial differential equations and by discrete cellular automata. Here we describe a simple three-states cellular automaton model based on the properties of excitation and recovery that are essential to excitable media. Our model is able to reproduce the dynamics of patterns observed in excitable media.

scholarly journals FORMATION OF SINGULARITY AND SMOOTH WAVE PROPAGATION FOR THE NON-ISENTROPIC COMPRESSIBLE EULER EQUATIONS

2011 ◽  
Vol 08 (04) ◽  
pp. 671-690 ◽  
Author(s):  
GENG CHEN
Keyword(s):  
Shock Wave ◽  
Wave Propagation ◽  
Initial Data ◽  
Differential Equations ◽  
Euler Equations ◽  
Hyperbolic Systems ◽  
Ideal Gas ◽  
Compressible Euler Equations ◽  
Compressible Euler ◽  
Shock Wave Formation

We define the notion of compressive and rarefactive waves and derive the differential equations describing smooth wave steepening for the compressible Euler equations with a varying entropy profile and general pressure laws. Using these differential equations, we directly generalize Lax's singularity (shock wave) formation results (established in 1964 for hyperbolic systems with two variables) to the 3 × 3 compressible Euler equations for a polytropic ideal gas. Our results are valid globally without restriction on the size of the variation of initial data.

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