The words waived, waved sound the same but have different meanings and spellings. Why do waived, waved sound the same even though they are completely different words?
The answer is simple: waived, waved are homophones of the English language.
Simple past tense and past participle of waive.
Simple past tense and past participle of wave.
Definitions from Wiktionary, Creative Commons Attribution/Share-Alike License and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.