KC-135A Water Injection Takeoff, from Martin Pole, with photographer's permission.
Here's the theory: Water is normally injected either at the compressor inlet or in the diffuser just before the combustion chambers. Adding water increases the mass being accelerated out of the engine, increasing thrust, but it also serves to cool the turbines. Since temperature is normally the limiting factor in turbine engine performance at low altitudes, the cooling effect allows the engine to be run at higher RPM with more fuel injected and more thrust created without overheating. The drawback of the system is that injecting water quenches the flame in the combustion chambers somewhat, as there is no way to cool the engine parts without cooling the flame accidentally. This leads to unburned fuel out the exhaust and a characteristic trail of black smoke.
Here's the reality: Materials used to make turbine blades surpassed temperature limits early on and by the early sixties engines that could produce more thrust without resorting to water injection were common, cheaper to operate, and easier to maintain.
I flew water injected KC-135A tankers for two years and it was a pain. See Flight Lessons / Leper Crew for a short story about water injection woes.
Figure: KC-135A Water Injection System, from KC-135A Flight Manual, figure 1-7.
[KC-135A Flight Manual, pg. 1-7] A water injection system provides thrust augmentation by allowing water to be sprayed into the air inlet and diffuser section of each engine. Water injected in this manner serves to increase the density of inlet and combustion air allowing increased thrust.
In the case of the KC-135A, 5,000 lbs of heated water boosted the thrust of each engine from around 9,000 to 12,000 lbs of thrust. The plumbing was complicated and prone to failure.
Portions of this page can be found in the book Flight Lessons 1: Basic Flight, Chapter 27.
Technical Order 1C-135(K)A-1, KC-135A Flight Manual, USAF Series, 25 April 1957