It is said that the gravitational waves are a consequence of the theory of relativity, of Albert Einstein, but it is difficult to find texts in which they argue why this relationship, except that they are arising from the study of the Einstein field equations. Let’s try to apply a little logic and some knowledge of relativity in this issue.
The key question is the following question: is possible to convey the effect of gravity faster than light?
Let’s assume for a moment that that is possible. For simplicity suppose that gravity affects instantaneously, or at infinite speed. That has a very important consequence, that is that you could create an experiment to transmit information instantaneously through instantaneous changes of gravitational effects. This break the relativity of simultaneity and would be able to define a simultaneity “objective”, that would lead us to be able to determine the “absolute space of reference”. The theory of relativity no longer would be valid from the point of view of affirming that all inertial reference systems are equivalent and that we cannot differentiate between them in any way. We could determine the existence of a privileged frame of reference.
But if the principle of relativity is valid… then by reduction to the absurd, we must think that gravitational effects can not be transmitted faster than the light. Indeed, no effect or anything can be transmitted faster than light.
Thus we have that the gravitational effects are transmitted at a finite speed which does not exceed at the light, and probably is transmitted at a rate equal to that of light.
So, a sudden change of mass or a mass movement will cause a gravitational shift at a point which will be broadcasted on the space at the speed of light. Thus gravitational waves arise. At the end of 1916 Einstein shows that the field equations also admit solutions in the form of waves. They are gravitational waves. 
For example if two stars are turning on themselves at high speed and at a distance not far from our solar system, the slight changes in the gravitational field that we perceive in our solar system should be received with a lapse of time of difference, for example, in the Earth than in Jupiter, and even a few miliseconds of difference between one point and one other of the world, and you could create an experiment that detected it.
By means of artificial satellites it has been able to detect small changes in distance between the satellite and the Earth that can be attributed to gravitational waves, but we still have to wait to get truly conclusive results by means of experiments of this kind.
In 1974 was detected a double pulsar whose observation provided interesting data for relativity . Its periapsis is moving about four degrees per year, and in addition the orbit of the star is shrinking in spiral and its period decreases. This shows a loss of energy which is attributed to intense gravitational waves.
New experiments are designed to detect gravitational waves as the LISA.