?H
” i
18
06
P
04
!J2 I
I
-25
30 i -- T - --2G
-
Fit,. 3. Solution of the telegraph equation with an origin-centered Gaussian function as
initidl condition. The solid cunr corresponds roughly to the configuration at the bottom of
Fig. 2 (and the initial distribution is a b u t as broad as one of the peaks here]; the dushed curve
corresponds to a more nearly difusive (kss wavelike) tekgraph equation.
any finite system is finite, so there are no infinitegpeeds (even without
invoking relativistic questions). The normal (Gaussian) solution for a point
source in molecular diflusion is, of course, just an approximation, albeit a
good one for times larger than a few mean free times. Appropriate generalizations for photon and molecular diffusion and Brownian motion started
with Fock (1926). A brief historical outline is given by Monin and Yaglom
(1965, 1971).
A second sensc in which turbuknt diffusion is “wavelike” for small times
is Ihitt the concentration field propagates dong essentially straight lines.
This hitppcns simply because fluid particle paths are differentiable at least
I wicc (the accelctations are finite). Therefore. neglecting simultaneous
nwlccular diffusion, the “characteristics “ of the relevant partial differential
equation. I-, + u Vl- = 0, are straight (in each “realization“ of a “Gibb\iaii ensemble”), o trait of simple wave phenomena. In his modeling of
turbulent diffusion by the hyperbolic tekgraph equation, Monin (1955,
1956) describes the systcm as displaying “diffusion with finite velocity“ (see
also Girgidov, 1973).
A third sense is basically the same as the second one: for t 4 0. the
” i
18
06
P
04
!J2 I
I
-25
30 i -- T - --2G
-
Fit,. 3. Solution of the telegraph equation with an origin-centered Gaussian function as
initidl condition. The solid cunr corresponds roughly to the configuration at the bottom of
Fig. 2 (and the initial distribution is a b u t as broad as one of the peaks here]; the dushed curve
corresponds to a more nearly difusive (kss wavelike) tekgraph equation.
any finite system is finite, so there are no infinitegpeeds (even without
invoking relativistic questions). The normal (Gaussian) solution for a point
source in molecular diflusion is, of course, just an approximation, albeit a
good one for times larger than a few mean free times. Appropriate generalizations for photon and molecular diffusion and Brownian motion started
with Fock (1926). A brief historical outline is given by Monin and Yaglom
(1965, 1971).
A second sensc in which turbuknt diffusion is “wavelike” for small times
is Ihitt the concentration field propagates dong essentially straight lines.
This hitppcns simply because fluid particle paths are differentiable at least
I wicc (the accelctations are finite). Therefore. neglecting simultaneous
nwlccular diffusion, the “characteristics “ of the relevant partial differential
equation. I-, + u Vl- = 0, are straight (in each “realization“ of a “Gibb\iaii ensemble”), o trait of simple wave phenomena. In his modeling of
turbulent diffusion by the hyperbolic tekgraph equation, Monin (1955,
1956) describes the systcm as displaying “diffusion with finite velocity“ (see
also Girgidov, 1973).
A third sense is basically the same as the second one: for t 4 0. the