When my Linux laptop is on BATTERY power, it makes soft low beeps
constantly.  These beeps are a mysterious annoyance.  Their volume is
not  affected by the laptop's volume-control or mute keys.  And they
only occur on Linux and on battery power.  (Note: The SpeedStep
feature reduces CPU clock
speed when I switch to battery power, so that could be involved.)

As far as I can tell, the beeps happen whenever there is a window
system event.  (For example, as I drag a window, there is a long
series of beeps.  Certain actions in Emacs also beep - I presume they
are repainting the window contents.)

I am running Red Hat Linux 7.3, GNOME, with the Sawmill window
manager.  I used to run Red Hat 6.2 on this laptop and didn't have the
problem, nor do I have it when I boot the laptop to Win 2K.  The
laptop is an IBM Thinkpad T21.

Thanks for any advice that helps me diagnose and fix this problem! 
The answer that leads me to the solution gets the payment.  (Yes, I'll
do my part to try to  track things down on the basis of your advice.)

I hate to see your question lying dormant, so I'll have a couple of
shots in the dark.

I presume you're using X, so have you tried typing the following in a
terminal:

xset b off

It should force X not to use the system speaker.

The other thing I thought of, and this is indeed a long shot, was that
your video drivers may in some way be interfering (radio noise) with
your sound card. How about dropping your resolution to see if the
problem persists? (As always, be careful if manually editing your
XF86Config). I remember way back in the day, I had a Cirrus Logic 1mb
card, and when I dragged a window, I would get a high pitched clicky
whine through my speakers if I was running at 1024x768@87hz
interlaced.

Good luck!

i believe seizer-ga has it right.  I get the same beeps on my laptop
and i have had it on past machines also.  It happened in both windows
and linux when i dragged a window or did anything involving graphics. 
infact my current laptop does it with any heavy network traffic but it
is very faint.  I have never found the exact reason but i would say
intereference is it.  If that is the case then there's really nothing
you can do about it as far as laptops go.