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Re: Standing waves

Quoting Ludwik Kowalski <kowalskil@MAIL.MONTCLAIR.EDU>:

The power formula in a textbook is derived
by integrating kinetic energy along one
lambda and then potential energy along one
lambda. The two are added and the sum is
divided by the period T. This gives

P=0.5*mu*w^2*A^2*v [1]

This is an object lesson in the perils of plug-and-chug,
or to say the same thing in more positive terms, a lesson
in the importance of knowing the domain of applicability
and validity of a formula.

That derivation and the resulting formula [1] make some
sense in the context of a running wave. They make no sense
whatsoever in the context of a standing wave.

The formula contains no parameters representing dissipation,
so it cannot describe a dissipative situation. In the
absence of dissipation, a standing wave requires ZERO
power, so the formula overestimates the power by a factor
of infinity.

In more detail: the formula [1] is off by a factor of Q,
roughly speaking. The formula tells you how much power you
put into the running wave, while Q tells you how many copies
of the running wave get reflected back to make the standing
wave you observe.