Propagation of Energy Pulses in Absorbing/Viscous Material Media
Abstract
A delta-pulse that emerges from the origin of coordinates in a lossy, viscous, homogenes medium is studied as it propagates through the medium experiencing absorption and dispersion. Elementary models for such propagation problems have appeared in classical textbooks (e.g., J.D. Jackson, Classical Electrodynamics, pp. 212-215, Wiley, 1962). Generally speaking, as the pulse propagates, its amplitude decreases, and its width broadens. A more detailed, exact analysis of this problem emerges from the study presented here in which the pulse obeys a third-order partial differential equation of parabolic type in time and one space dimension, first obtained by Stokes in 1885. We obtain the exact solution of the pertinent boundary and initial value problem (BIVP) posed in a rigorous fashion, in which the initial displacement is the delta-pulse in question. The resulting exact distortion of the pulse (i.e., amplitude-decaying and width-broadening shape) emerges from a series solution which we have obtained by Laplace transform techniques. The solution exhibits the expected smoothing-out effects of dispersion. A portion of the final expression, which contains a sum of repeated integrals of the complementary error function, is recast in power series form, thus simplifying the final result. We also show how to obtain an approximation agrees with the leading term of the exact solution described above. Further examination of the dispersion relation associated with the governing PDE shows that the kinematics viscosity of the medium must be related to the pulse propagation speed in a specific way to insure that propagation with attenuation actually occurs. Under this simple but restrictive condition, quantitative details of the distorted pulse amplitude are illustrated in several nondimensional graphs. These are plotted versus time (or position) for discrete successive values of the spatial position (or time).
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 2000
- Accession Number
- ADA392375
Entities
People
- G. C. Everstine
- G. C. Gaunaurd
Organizations
- Naval Surface Warfare Center Carderock Division