I have an idea for designing a light, cheap and fast 2-seated private airplane... I was thinking about using a pulse jet engine which is simple to design and build, and is cheap too.
Noise is not an important concern in my region. The only things that I'm wondering about are efficiency, fuel consumption (range), vibrations and cooling.
Is it feasible to use pulse jet instead of regular engines?
Is the pulse jet functional and operational?
In principle yes. But some details will turn most potential operators off.
Noise is the obvious first one. You might not mind, but cockpit noise and vibration in operation will certainly put an unacceptable stress on the pilot. Just think how you will communicate with ATC or with a co-pilot. Also, your choice of airports to operate from will be rather restricted.
Vibration also means that you need to ruggedise the airframe. This will certainly eat up some of the mass benefits of the pulse jet, but is hard to quantify.
Now fuel consumption: The AS-014 would consume 350 g of fuel per hour for every Newton of thrust. Compare this to the 60 g/Nh of a modern turbofan. The additional fuel will eat up the remaining mass advantage once you desire to fly for more than just a few minutes.
Next will be take-off distance. Unless you use a catapult, you will need a long runway. The pulse jet has limited static thrust, so your initial acceleration will be poor. At the high end of the speed envelope, though, thrust should be plentiful.
Cooling, however, is no problem. Just expose the pulse jet to the elements and you are all set.
To summarise: Yes, a pulsejet is a possible power source for a GA airplane, but is a poor choice for a multitude of reasons. It makes more sense in an unmanned, one-way application.
This question is pretty close.
And you should read here as well as here
Generally pulse jets are not great at low speeds (and can be hard to start when stationary). These are key aspects of small GA planes. The noise may not be an issue for you locally (in terms of noise abatements) but it may be an issue for the occupants of the aircraft.
In short it is possible (physically) but not really practical.
While the pulsejet powered Kawanishi Baika from the WW2 was really far from the GA, from the picture it looks like it was a relatively small plane of very comparable size, capable of taking up under own power, manned. While none were eventually built, seems that the idea was accepted rather seriously at the time. (Its sister ship is the Fieseler Fi 103R, of which 175 were built and a few were test-flown. But they launched by dropping from a bomber, which avoided the ground-launch disadvantages noted in Peter Kämpf's answer. Again, not practical for light GA.)
