I have studied only some basic stuff on mechanical waves
Consider a string attached to wall at one side and you're holding the other side so that the string is taut. What I've noticed is that when the source is moving up and down in SHM, then period of source is same as period of wave traveling. I don't know why, but this is really difficult for me to grasp intuitively as when I think about this, I end messing up everything. So, I would like to have your view on this and want to know how you think about this seemingly strange fact.
Also, I want to know whether this is true in general in a situation wherein the source is moving periodically (but not necessarily simple harmonically) and time periods of the source and the wave are exactly the same. A mathematical treatment of this would be helpful, if that is not too advanced.
Please help me on this.