Wave is a disturbance or energy that propagate through medium from one point to other point
So basically it is a flowing energy that flows into the medium and hence medium particles start oscillating about their mean position to and fro.
This motion of medium particles or this to and fro motion is about their mean position and this will always be cyclic or periodic motion
This means the disturbance or energy continuously flow through the medium such that it will change the position of medium particle and this will be cyclic in order
For an example
so here above equation of wave is a travelling wave in which displacement of medium particle from its mean position is given by "y"
Now we can see that this disturbance depends upon the sine function and it will repeat its same position after every 2 pi time interval as it is cyclic function for this value
Due to this phenomenon of repeatation of its same position we can say that it is disturbance of wave is cyclic.