Re: Energy Transmission on a string.

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding John Denker's response to Joel's comment:

> ...     We are !!not!! assuming sine waves
> here.  It's pretty clear that the energy moves along with the
> wave packet, ....
>
> > Would it be fair to interpret what you are saying here as implying
> > that it is probably mis-placed mathematical rigour on the part of
> > introductory authors, who use method B, to arrive at the standard
> > result
> >
> > 1/2 *mu * v *omega^2 * A^2
>
> Absolutely that is not a rigorous result.  In the real world
> there are all sorts of nonlinearities, and even in a linear
> system there is usually dispersion that calls into question
> the very notion of a definite speed of propagation.

Not only is Joel's above formula restricted to linear sine waves,
etc., it is actually a *spatio-temporal average* (over an integer
number of complete oscillations) of the propagated power along the
string even when such other restrictions apply.

It seems to me that we can do a little better.  For instance, if we
*do* make the concession that we are restricting ourselves to
propagation on a string that neglects string stiffness and other
nonlinear effects, and makes the small (maximum) angle approximation,
but do *not* make any assumptions about the spatio-temporal waveform
(other than the small slope restriction applies), and in particular
do not average over time or space--keeping everything instantaneously
local, we get the alternative form below for the locally (in time and
in space) propagated power:

P = -T*v_t*m

where T is the string tension, v_t is the instantaneous local
transverse velocity of an element of the string, and m is the
instantaneous local slope of that element.  The reason for the
negative sign is that a motion that has a positive-sloped piece of
the string moving upward (or a negative-sloped piece moving downward)
will have the local energy flux propagate to the left, and any
negative-sloped piece of the string moving upward (or a positive-
sloped piece moving downward) will have the local energy flux
propagate to the right.  (Note that nowhere does the actual phase
and/or group velocity of the waves appear in the formula.)

In the special case of a running wave we can observe that this
formula results in energy flowing in the same direction as the
running wave propagates.  But in the case of a standing wave it
results in an energy flux that oscillates in direction with time so
over a complete cycle there is no net longitudinal energy transport
along the string anywhere.

Note that even though the direction of longitudinal energy transport
is always in the same as the direction of the wave motion in the case
of a running wave, the local magnitude of this flux is space/time
dependent.  The flux is a maximum at the regions of the local zero
crossings of the displacement wave, and it is minimum (actually
zero) at the tops of the crests and bottoms of the troughs.  The
maximum value at the zero crossings is twice that given by Joel's
averaged formula, but when this is averaged over space (and/or time)
over a complete cycle, Joel's average value obtains in the end.

David Bowman
David_Bowman@georgetowncollege.edu