1. To move up and down, back and forth freely or haphazardly. The flag waved around wildly in the wind. The tower started waving around unsteadily, but thankfully it didn't collapse.
2. To move or swing something up and down, back and forth in the air. In this usage, a noun or pronoun can be used between "wave" and "around." Please don't wave that stick around—you might break something! The protester stood aloft the barricade and began waving around the banner of the resistance.
3. To signal for someone or a vehicle to move around someone or something. In this usage, a noun or pronoun can be used between "wave" and "around." The cop waved everyone around the paramedics as they treated the injured man. We need someone to stand in front of the roadworks and wave around the cars in traffic.
4. To display, promote, or show off someone or something, especially in a superficial, self-serving, or self-important manner. In this usage, a noun or pronoun can be used between "wave" and "around." The only recognition our team ever gets is when the boss waves us around at the investors' meeting each year. I don't want to spend all that time and money just to get some degree that I can wave around but doesn't lead to a lucrative career. The federal agent came in and started waving her badge around, demanding to be shown our financial accounts.
wave something around
to raise something up and move it around so that everyone can see it. When Ruth found the money, she waved it around so everyone could see it. She kept waving around the dollar she found in the street.