For light, we all know that the energy-momentum relation is E=cp. Reuben
shows that EXACTLY the same relationship holds for mechanical waves, where
E and p are now average (over either time or space) densities (eg. per unit
length for a wave on a string, per unit volume for sound, etc.) and c is
the wave speed.
So the answer to my original question is: all traveling waves do carry a
nonzero average momentum. Hence, the Maxell ad, although wrong in
magnitude, is right in principle: a sound wave will blow your hair back.
You can calculate from the above relationship that the force should be
experimentally verifiable for a decent stereo system - ie., I suggest
dropping a feather in front of your speakers when the volume is cranked and
there are otherwise no drafts in your room and see if there is a net drift.
I haven't tried the experiment myself, but I'm dying to do so.
This also means that a wave on a string is not simply transverse. There is
a forward component. I think it's correct to think of this as the symmetry
breaking phenomenon which drives the wave forward rather than backward,
though I'd welcome comments on this interpretation.
I also think we can carry Reuben's analysis a step further. Let's write
p=mc where m is the net mass transmitted by the wave. Hence we have E=mc^2.
I think the reason this didn't come out as 0.5mc^2 is because the KE and PE
of any unit length (or volume) of the displaced medium are equal so that
E=2*KE. I would then conclude that typical textbooks which state that there
is no bulk mass flow for a wave on a string, water surface, sound, etc are
WRONG. (Of course, the wave must be traveling not standing, I hasten to
add.) After all, this is what I usually do to elongate my garden hose when
I'm too lazy to walk to the other end and tug on it.
Dr. Carl E. Mungan, Assistant Professor http://www.uwf.edu/~cmungan/
Dept. of Physics, University of West Florida, Pensacola, FL 32514-5751
office: 850-474-2645 (secretary -2267, FAX -3323) email: cmungan@uwf.edu