To waive is (1) to give up a claim or right voluntarily, (2) to refrain from enforcing something, or (3) to set something aside temporarily. To wave is to move back and forth or up and down or to make an up-and-down or back-and-forth hand signal. The words are closely related to waiver and waver, which have related distinctions.
Examples
The guard will wave you through, and you will follow the road around the entire stadium until you come upon Parking Lot P. [Los Angeles Times]
Some insurers will waive the excess if you make a claim in Europe, but have used an EHIC card. [Telegraph]
They could have waved to each other as they crossed paths on their way to their new teams. [Stuff.co.nz]
Currently, visitors from Canada, Mexico, and some Caribbean countries have been waived from paying a fee. [CTV]