chip@t4test.UUCP (Chip Rosenthal) (09/23/84)
(This is a pretty long article whereby I hopefully derive my final
conclusion correctly. If you find this whole derivation business
boring, the meat of the discussion is really the "depth of penetration"
information at the end.)
A transverse electromagnetic plane wave traveling in the x
direction can be described by:
E(x) = E exp[ -gx ]
0
where gamma (g) is a complex number called the `propagation constant'.
Gamma depends upon two things, the properties of the medium
(specifically permetivity, permeability, and conductivity) and the
frequency of the wave. To make life easier, we are assuming an
isotropic medium, which means directions and locations don't effect the
medium's properties. By the way, if you are wondering where time comes
in:
E(x,t) = E(x) * exp[ -jwt ]
where:
w = angular frequency (omega)
j = sqrt(-1)
If you play around with Maxwell's equations, you can get an identity for
gamma as follows:
2 2 js
g = [ -w ue * ( 1 - -- ) ]
we
where:
u = permetivity of the medium (mu)
e = permeability of the medium (epsilon)
s = conductivity of the medium (sigma)
Now take a look at gamma for two cases: a good conductor (i.e. the
conductivity term is much greater than one) and a good dialectric
(i.e. the conductivity term is much less than one). For a good
conductor, you get:
2 wus wus
g = -jwus --> g = sqrt( --- ) + j*sqrt( --- )
2 2
Notice that the propagation constant has a real and an imaginary term.
Since the real term gives a negative exponential in the equation for
the wave, the wave dies exponentially as it travels in the medium.
Now, for a dialectric, gamma becomes:
2 2
g = -w ue --> g = jw*sqrt(ue)
Notice that this is a purly imaginary term. The wave will propagate
without dissipation. (Actually, a better assumption might have been
that the imaginary term is dominated by w*sqrt(ue), with a very small
real term still present. But what the heck, I'm making the arbitrary
assumptions here.)
Getting back to the original question. Where do waves propagate
better: salt water or fresh water? Well, salt water is a better
conductor than fresh water, so it will have a larger real term in
the propagation constant. Therefore, waves propagate better in
fresh water than salt water.
One last tidbit I want to through out is the "depth of penetration".
This is analogous to the time constant for an RC circuit; it tells
you where the wave has attenuated to 1/exp (approx 37%). Without
going through the math, the answer is:
1 2
d = ------ = sqrt( --- )
REAL{g} wus
where:
d is the depth of penetration (delta)
Not only does the depth of penetration decrease as the conductivity
increases, but it also increases as the frequency increases. Which
leads us back to the beginning of this whole discussion: why do you
need extremely low frequency waves to get through water.
I went looking for some numbers to plug into the equation, and instead
found an interesting graph. It is figure 7-8 of "Classical
Electrodynamics" by Jackson. If I pull some numbers out of the graph
and put them in terms of the quantities discussed above, I get:
depth of penetration
frequency salt water fresh water
100Hz 10m
10KHz 1m
1MHz 0.1m 10m
10MHz 0.01m 10m
1GHz 0.01m
The graph shows sea water only from 100Hz to 10MHz. It is a
straight line on this log/log graph. Fresh water is flat up to
10MHz, shoots up fast to 1THz, and then does funny things in the
region of visible light.
(Hah! Will the folks at cornell!tesla tell Prof. Kelly that I
must have gotten something out of his course?)
--
Chip Rosenthal, Intel/Santa Clara
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