Compare these two examples, but with the verb "to throw". I threw my book to him. I threw my book at him. Would that make the difference any clearer? (I normally demonstrate this for real in class.)

Hi Vika! How have you been? According to what I learned from my friends and relatives from the US, when you wave to someone, you are simply responding to a person who also waved to you. Now, waving at someone can have a broader meaning e.g, raising your hand as to signal recognition to someone. Example: Ross waved at Rachel as soon as he went out of the airpot gates. I hope this will do. Cheers! :)