The words waists, wastes sound the same but have different meanings and spellings. Why do waists, wastes sound the same even though they are completely different words?
The answer is simple: waists, wastes are homophones of the English language.
Plural form of waist.
Plural form of waste.
Third-person singular simple present indicative form of waste.
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").