The words waive, wave sound the same but have different meanings and spellings. Why do waive, wave sound the same even though they are completely different words?
The answer is simple: waive, wave are homophones of the English language.
To give up (a claim or right) voluntarily; relinquish. See Synonyms at relinquish.
To refrain from insisting on or enforcing (a rule or penalty, for example); dispense with: "The original ban on private trading had long since been waived” ( William L. Schurz).
To put aside or off temporarily; defer.
To move freely back and forth or up and down in the air, as branches in the wind.
To make a signal with an up-and-down or back-and-forth movement of the hand or an object held in the hand: waved as she drove by.
To have an undulating or wavy form; curve or curl: Her hair waves naturally.
To cause to move back and forth or up and down, either once or repeatedly: She waved a fan before her face.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.