No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

The full text of this article is not currently available.

Equations for the propagation of finite local disturbances in vector fields of physics and mechanics

### Abstract

The purpose of this theoretical investigation is to determine the propagation velocities of local disturbances that may occur in the curl and divergence of a vector field v that satisfies the equation ∂v/∂*t*+f=0, where f is a general vector field.Equations of this form frequently occur in physics and mechanics. A *d* *i* *s* *t* *u* *r* *b* *a* *n* *c* *e* in the curl v (div v) field is defined by ∂curl v/∂*t*≠0 (∂div v/∂*t*≠0). Such a disturbance is termed a *l* *o* *c* *a* *l* disturbance if it is surrounded by a quiescent layer of finite thickness. Equations are derived for the velocity U(r,*t*) of curl v lines and the velocity W(r,*t*) of div v elements. These velocities conserve, respectively, the circulation of v about arbitrary circuits and the efflux of v through arbitrary closed surfaces. These conservation conditions are not sufficient to completely specify the two velocity fields, and each contains an arbitrary term. More convenient for three‐dimensional applications than the circulation is the gyration G (spatial integral of curl v); and U is found to also conserve G in regions T where the curl v lines retain their individual shapes and orientations as they move, and our consideration of curl v disturbances is confined to such regions. The centroid of efflux is defined for arbitrary volumes, and the centroid of gyration for volumes in T. The movement of these centroids is linked to the velocities U and W by the conservation relations. Contributions from the arbitrary term in U (W) are suppressed by integration over a *l* *o* *c* *a* *l* curl v (div v) disturbance and by construction of a scalar potential field that satisfies Laplace’s equation in the interior of the disturbance with Dirichlet (Neumann) conditions related to f on the closed boundary. Formulas for the velocities of the centroids are thus obtained for local curl v and div v disturbances as functions of f, and these velocities are interpreted as propagation velocities of the disturbances. Only one type of disturbance need be localized for its formula to apply. These formulas enable us to calculate how such local disturbances will propagate without integrating forward in time; and in certain cases, only the fields outside a disturbance are required for this calculation. Applications to Maxwell’s electrodynamic equations are presented as examples.

© 1979 American Institute of Physics

Published online 29 July 2008

/content/aip/journal/jmp/20/7/10.1063/1.524227

http://aip.metastore.ingenta.com/content/aip/journal/jmp/20/7/10.1063/1.524227

Article metrics loading...

/content/aip/journal/jmp/20/7/10.1063/1.524227

2008-07-29

2016-09-29

Full text loading...

###
Most read this month

Article

content/aip/journal/jmp

Journal

5

3

Commenting has been disabled for this content