Can gravitational wave create anti-gravity, i.e. repulsive gravity? A very layman question as in title. Like every wave having a negative side, can a gravitational wave have anti-gravity.
To put it in different words, a gravitational wave passing through a complete vacuum, if in positive cycle, can create a denser space-time, in it's negative cycle, create a rarer space-time?
 A: Gravitational waves, though transverse, can be thought of as similar to sound waves: 
A sound wave, as it moves through a medium the sound wave creates alternating volumes of greater and lesser particle density.
Gravitational waves do something similar, except the medium is spacetime itself.  The result is that as a gravitational wave passes through a region of space, at one crest the spacetime is "stretched" in one direction and contracted in the perpendicular direction, like when you stretch a rubber band and it gets narrower. At the trough of the wave, the same thing happens, except the direction that was contracted is now stretched and the direction that was stretch is now contracted. This is why the good ol' perpendicular lasers and mirrors trick worked for detecting them.
A: Gravitational waves are not cycles of compression and rarefication like sound waves. They're transverse, and there is no such thing as compression or expansion of spacetime. There is curvature of spacetime. In a gravitational wave, the curvature is what oscillates.
In general relativity, the precise definition of what we mean by attractive or repulsive gravity is complicated, and difficult to express without some mathematics. We express this definition in a set of various criteria called energy conditions. The energy conditions are all automatically obeyed in a vacuum, so gravitational waves do not contain repulsive gravity.
A: Gravity is always attractive as it creates a positive curvature in spacetime. Then, such curvature in the 4D hypersurface is like "a mesh" of geodesics. These geodesics are the path you follow in a free fall when attracted by another massive body nearby, made of positive mass. The local curvature of the 4D hypersurface is then locally positive (spherical geometry), and the geodesics always evoke gravitational attraction.
If one could generate a negative curvature in spacetime, either by producing some exotic matter of negative mass, or by concentrating enough negative energy density locally (neither is known), then a negative curvature would be induced in spacetime, a hyperbolic geometry (or horse saddle geometry) producing geodesics and a free-fall path that evoke gravitational repulsion, i.e. antigravity.
As for the gravitational waves, they have indeed been detected (by LIGO/Virgo), but their associated quantum that would mediate gravity, a 2-spin boson called the "graviton", is still hypothetical and AWOL for a century. Not all fields have a mediate elementary particle. Gravitational waves could just be "ripples in spacetime", like 2D ripples on the water surface of a pond. Einstein's coupling constant 8πG/c⁴ in his field equations tells us how is such "coupling" i.e. the direct relation between how much spacetime is distorted (the Gμν tensor, LHS) with respect to the amount of local matter-energy (Tμν tensor, RHS). It shows the stiffness of spacetime and how it is extremely rigid: indeed, the coupling constant has the light speed raised to the fourth power at the denominator, making it extremely small.
So any gravitational waves are very tiny, even with a lot of energy focused locally, so any gravitational waves that would be artificially produced in the future are expected to be very weak. It is no coincidence that such waves have been detected thanks to the gigantic energy released by the merging of two black holes, yet the signal was almost lost in the noise and its detection required a particularly elaborate algorithm.
A: I will re-form your question by ignoring the second and third sentence, which are not too clear. 
Can gravitational wave create anti-gravity, i.e. repulsive gravity?
Yes.
Its the same as for light: if light - which carries momentum - is absorbed by an object then that object moves away from the light source.
The trick with gravitational waves is that normal matter does not absorb gravitational waves very well. (But there is always some absorption). 
To Maximize the Effect:
If gravitational waves impinge on any rotating object there can be repulsion or attraction. The effect is only really strong when waves impinge on a rapidly rotating compact object like a spinning black hole. To get a nice large effect the period of the waves need to be of the same size as the spin rate. 
The effect when its an near coherence mode is called super radiance. 
For repulsion the effect is actually an absorption of gravitational energy from the wave, so that the object starts to move in the same direction as the wave.  See Figure 16 of Brito: - http://arxiv.org/pdf/1501.06570v3.pdf
The effect can be quite pronounced. 10% of the incoming energy of the wave can be absorbed. 
A: So it looks:

Some more details:


*

*The GWs are transverse waves, not longitudinal ones

*Gravity is not a force in the GR

*They don't produce "anti-gravity", i.e. their effect isn't in an opposite direction to the source, it is perprendicular to them

*Also the GWs don't produce a force, they change the metric in the space they travel. As this change shrinks in a direction, it grows in the other, thus the volume they travel doesn't change. Like so (source):



