The words waiving, waving sound the same but have different meanings and spellings. Why do waiving, waving sound the same even though they are completely different words?
The answer is simple: waiving, waving are homophones of the English language.
Present participle of waive.
Present participle of wave.
Repeated moving of arms or hands to signal.
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.
If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").