WE HAVE A LONG RUN FROM OUR WATER HEATER TO A SHOWER ABOUT 35 FEET.
THE SHOWER IS FEED BY A 3/4" COPPER LINE. I'M THINKING OF CHANGING THE
WATER LINE TO 3/8 O.D. LINE. MY THINKING IS THAT I WOULD GET HOT WATER
A LOT QUICKER BECAUSE WHEN YOU FIRST TURN ON THE SHOWER IT WOULD NOT
HAVE TO DISPLACE AS MUCH COLD WATER IN A 3/8" LINE VERSUS THE 3/4"
LINE. WITH A SMALL HOLE IN THE SHOWER HEAD THIS RESTRICTS THE FLOW OF
WATER WHICH TAKES PLENTY OF TIME WITH A 3/4" LINE.

ONLY TWO PEOPLE LIVE IN THE HOUSE SO WE USUALLY HAVE TO BLEED OFF ALL
THE COLD WATER EVERY TIME.

HOW MUCH FASTER WILL I GET HOT WATER?

THANKS, R. MEADE

Hi, blackbob-ga:

The smaller line will have less volume, and hence less cold water to
displace, but it will also have a smaller cross-section and hence
restrict the rate of flow more than the larger line does.

Note that reducing the diameter by half cuts the cross-section area by
1/4th.

I think the slower flow pretty much offsets the reduced volume, so
you'd wind up waiting just as long.  Also, in my experience the
restricted flow from a 3/8ths inch line makes for a pretty
unsatisfying shower.  That, however, is a matter of personal taste.

regards, mathtalk-ga

Dear Blackbob and Mathtalk,

Just to clarify: reducing the diameter of your pipe by a half reduces
its cross-sectional area _by_ three-quarters - in other words, _to_
one quarter - of what it was, not by a quarter.  (Since both figures
are outside diameters, the reduction will be slightly more extreme
than this, of course.)

I trust this helps.

Carnegie

If the resistance offered by the pipe is negligible compared to the
resistance of the shower head, you'd only have to wait 1/4 as long. 
If the resistance of the shower head is negligible compared to that of
the pipe, on the other hand, you'd actually have to wait 4 times
LONGER, because flow in a pipe goes like the fourth power of the
radius.  With a 3/8" line, however, both effects are certainly
significant.
From a practical point of view, 3/8" pipe is almost never used for
water plumbing, and there's probably a reason.  1/2" is used all the
time, and shouldn't decrease the flow rate much, especially if there
aren't too many elbows.  Using 1/2" would still decrease the amount of
water to bleed off by more than a factor of 2.

by which I mean to less than half of it's former value

Hi, blackbob-ga:

A couple of alternative approaches:

[Hot Water Recirculation]
http://www.grundfos.com/Web/HomeUs.nsf/Webopslag/D6600B14F5C7C73386256AE9005326AE

The idea is to pump hot water that is sitting still in the pipe back
to the hot water heater periodically.  While this wastes energy (in
the form of heat that dissipates in the pipes more than otherwise), it
saves that cold water that would otherwise run down the shower drain
waiting for the hot water to arrive.

[Tankless Hot Water Heaters]
http://www.chilipepperapp.com/hwcs.htm

Actually this last site discusses the gamut of hot water recirculation
options in an attempt to convince you to go with a tankless hot water
heater.  These are small electric heaters that can be located close to
the point of use, and deliver a reasonable throughput of hot water "on
the fly".  This approach probably saves the most in waste water +
energy in the long run.

regards, mathtalk-ga