Re: Speed of light straight from the classical wave equation ?
- From: Randy Poe <poespam-trap@xxxxxxxxx>
- Date: Wed, 30 Jan 2008 06:14:00 -0800 (PST)
On Jan 30, 8:41 am, srp2...@xxxxxxxxx wrote:
Out of curiosity, I did a dimensional analysis of the very
straightforward wave equation
describing a moving transverse wave on an elastic string.
On the hyperphysics page,
v = sqrt[ T/(m/L) ]
where m is mass of string, T is tension, and L is
length of string.
http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/string.html
It would be more comprehensible to actually state
the equation you're talking about, or give a link.
you have mu which is in kg/m
Not necessarily, but I can imagine that there would
be an equation where the symbol mu was used to
represent mass per unit length.
A standard scalar wave equation is:
@^2 u/@t^2 = v^2 * del^2 u
where "@" indicates partial derivative, u is the quantity
which is oscillating, del^2 means the Laplacian, and
the parameter v turns out to be the propagation
speed.
For EM radiation, u is a field strength. For
sound waves, u is pressure. For your oscillating
string, u is transverse distance. All of these
things have different units.
you have T which is in Newton
you have (2nd p derivative on position) f = mu/T (2nd p derivative on
time)
I take it f is the symbol for displacement here.
So mu/T = 1/v^2, or v = sqrt( T/mu ) as I said above.
Now, from E=mc^2 mass is really E/c^2, which is Joules/c^2
Yes, Joules/c^2 are a mass unit. I don't know if
I'd say mass is "really E/c^2". But that's one
possible system of units.
which gives mu the dimensions Joules/(c^2 m)
Newton is, after resolving to its most fundamental dimensions, Joules/
meter
What does "most fundamental" mean? Yes, a Newton
is a Joule/meter. It's also a kg*m/sec^2, which
is probably more fundamental since those are the
fundamental mks units.
which results in the dimensions of mu/T simplifying as follows
mu/T = [J/(c^2 m)] (m/J) = 1/c^2
Yes mu/T = 1/v^2 has the units of inverse velocity
squared.
Conclusion:
There is not even need to go to derivations from Faraday and
Maxwell's forth equation to get it.
No, there is no need to go to Faraday and Maxwell
to conclude that velocity of a wave on a string
has the units of velocity.
Nor does anything in Faraday or Maxwell have anything
to say about what the velocity on a string is.
I sort of wonder what the meaning of this is.
I wonder what you think "this" is.
You seem to have concluded that velocity has
units of velocity (already known), that mu/T has
units of inverse velocity-squared (already known,
standard equation for speed of waves on a string),
and that you don't need Maxwell's equations to
recognize what parameter represents speed in a
general wave equation (also already known).
- Randy
.
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