Internal diffusion in a biporous adsorbent with rectangular adsorption isotherm 4. Integral equations for adsorption wave front movement in spherical and cylindrical grains

1978 ◽  
Vol 27 (11) ◽  
pp. 2189-2193
Author(s):  
P. P. Zolotarev ◽  
V. I. Ulin
Keyword(s):  
Adsorption Isotherm ◽  
Wave Front ◽  
Integral Equations ◽  
Internal Diffusion ◽  
Adsorption Wave ◽  
Front Movement

THE STOCHASTIC NATURE OF CARDIAC PROPAGATION DUE TO THE DISCRETE CELLULAR STRUCTURE OF THE MYOCARDIUM

1996 ◽  
Vol 06 (09) ◽  
pp. 1637-1656 ◽  
Author(s):  
MADISON S. SPACH
Keyword(s):  
Gap Junctions ◽  
Wave Front ◽  
Cellular Structure ◽  
Conduction Block ◽  
Microscopic Level ◽  
Cardiac Conduction ◽  
Electrical Load ◽  
Stochastic Nature ◽  
Myocardial Architecture ◽  
Front Movement

The object of this paper is to describe cardiac conduction phenomena caused by the discrete nature of cardiac cellular structure. Recent results show that the myocardial architecture creates inhomogeneities of electrical load at the microscopic level that cause cardiac propagation to be stochastic in nature. That is, the excitatory events during propagation are constantly changing and disorderly in the sense of varying intracellular events and delays between cells. A unique feature of the stochastic nature of cardiac propagation is that electrical boundaries produced by cellular myocardial architecture create inhomogeneities of electrical load that affect conduction inside individual cells and influence conduction delays across gap junctions, as well as at muscle bundle junctions. This process produces discontinuous propagation as a primary reflection of the nonuniformities of electrical load due to the irregular arrangement of the cellular borders and the associated nonuniform distribution of their electrical interconnections. A fundamental consequence of the stochastic nature of normal propagation at a microscopic level is that it provides a major protective effect against arrhythmias by re-establishing the general trend of wave front movement after small variations in excitation events occur. When the diversity at a very small size scale decreases throughout the tissue, such as occurs when there are regularly repeating relatively isolated groups of cells, larger fluctuations of load can develop and be distributed over more cells than occurs normally. The myocardial architecture may then fail to re-establish a smoothed wave front and re-entry can develop. These relatively new discontinuous conduction phenomena provide important theoretical and experimental challenges to synthesize a complete theory linking continuous and discontinuous media as applied to cardiac conduction. The results show that it will be important to distinguish differences in wave front movement and conduction block caused by mechanisms of continuous media versus wave front movement and block imposed by directional or localized changes in cellular connectivity; i.e., the topology of the electrical connections between cells (gap junctions).

On the Group Front and Group Velocity in a Dispersive Medium Upon Refraction From a Nondispersive Medium

2003 ◽  
Author(s):  
Z. M. Zhang ◽  
Keunhan Park
Keyword(s):  
Wave Packet ◽  
Phase Velocity ◽  
Wave Front ◽  
Group Velocity ◽  
Dispersive Medium ◽  
Front Velocity ◽  
Normal Group ◽  
Nondispersive Medium ◽  
Positive Index ◽  
Front Movement

Conventional definitions of the velocities associated with the propagation of the modulated wave are both confusing and insufficient to describe the behavior of the wave packet clearly in a multi-dimensional dispersive medium. There exist infinite solutions to the general equation of wave-front movement, suggesting that there are infinite phase velocities. Therefore, the introduction of “normal phase velocity” becomes necessary to unambiguously define the phase velocity in the direction perpendicular to the wave front. Similarly, there exist infinite solutions to the equation describing the group-front movement in the case of a wave packet, resulting in infinite “group-front velocities.” The “normal group-front velocity” is defined as the smallest speed at which the group-front travels and is in the direction perpendicular to the group front. We show that the group velocity (i.e., the velocity of energy flow) is one of the group-front velocities and, in general, is not the same as the normal group-front velocity. Hence, the direction in which the wave packet travels is not necessarily normal to the group front. Examples are used to demonstrate the behavior of a wave packet that is refracted from vacuum to a positive index medium (PIM) or a negative index medium (NIM).

Application of Kirchhoff’s Integral Equation Formulation to an Elastic Wave Scattering Problem

1967 ◽  
Vol 34 (4) ◽  
pp. 921-930 ◽  
Author(s):  
William L. Ko ◽  
Thorbjorn Karlsson
Keyword(s):  
Wave Front ◽  
Integral Equations ◽  
Scattering Problem ◽  
Linear Matrix ◽  
Infinite Length ◽  
Shadow Zone ◽  
Surface Integral ◽  
Front Solution ◽  
Frequency Wave ◽  
Integral Equation Formulation

Interaction of a plane compressional step wave with a circular cylindrical obstacle embedded in an elastic medium is studied. The obstacle is rigid, stationary, and of infinite length. The incident wave travels in a direction perpendicular to the axis of the cylinder. Using Kirchhoff’s theorem, surface integral equations are formulated for the displacement potential derivatives in the scattered field and on the cylinder boundary. The wave-front solution obtained for the illuminated zone on the cylinder is identical to that obtained by high-frequency wave-front analysis. Boundary stresses in the shadow zone as well as the initial behavior of the wave-front stresses at the boundary between the illuminated and shadow zones are obtained. The integral equations for both illuminated and shadow-zone boundary stresses are reduced to successive linear matrix equations for numerical analysis. The numerical methods developed in this paper can be applied to interaction problems for obstacles of arbitrary geometrical configuration. They are also readily extended to the case where the medium exhibits bilinear or multilinear stress-strain behavior.

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