waive, wave

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.

waive
  1. :: verb-transitive

    To give up (a claim or right) voluntarily; relinquish. See Synonyms at relinquish.

  2. :: verb-transitive

    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).

  3. :: verb-transitive

    To put aside or off temporarily; defer.

wave
  1. :: verb-intransitive

    To move freely back and forth or up and down in the air, as branches in the wind.

  2. :: verb-intransitive

    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.

  3. :: verb-intransitive

    To have an undulating or wavy form; curve or curl: Her hair waves naturally.

  4. :: verb-transitive

    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.

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About Homophones

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").