What does ~ mean in text?

[Aedistopholes]

Since you mentioned it specifically referencing the Japanese:
The wave dash (波ダッシュ nami dasshu?) is used for various purposes in Japanese, including to denote ranges of numbers, in place of dashes or brackets, and to indicate origin. The wave dash is also used to separate a title and a subtitle in the same line, as a colon is used in English.

When used in conversations via email or instant messenger it may be used as a sarcasm mark or, in East Asia, as an extension of the final syllable to produce the same effect as "whyyyyyy" with "why〜". Used at the end of a word or sentence in text communications, it often denotes something said in a sing-song or playful voice, or similar to the use in instant messengers and email, depending on context. In some contexts, the tilde represents a lustful or exhausted sigh: "Hello there~".