In modern Japanese, old /β̞u/ evolved into /u/, and is written as う in hiragana and ウ in katakana. Sound /β̞u/ reintroduced later is written as うぅ in hiragana and ウゥ in katakana - however most speakers pronounce this as a long /u/ sound.
In Meiji era grammar books, was sometimes used to spell ワ行下二段活用 verbs.
In ancient times, all は行 sounds were pronounced /pV/ (“V” being any of the Japanese vowels). Then, during the Nara period, it shifted to /ɸV/ when it was in the middle or at the end of a word. Between the Heian and Kamakura periods, /ɸV/ changed to /βV/→/β̞V/→/V/. Some older grammar books used to show this in ハ行四段活用 verbs, ハ行上二段活用 verbs, and ハ行下二段活用 verbs (e.g. 思ふ(omofu) as 思(omowu).
The katakana syllabary is used primarily for transcription of foreign language words into Japanese and the writing of gairaigo (loan words), as well as to represent onomatopoeias, technical and scientific terms, and the names of plants, animals, and minerals. It is also occasionally used in some words for emphasis, or to ease reading; katakana may be preferred for words becoming buried in the text if they are written under their canonical form in hiragana. Names of Japanese companies, as well as certain Japanese language words such as colloquial terms, are also sometimes written in katakana rather than the other systems. Formerly, female given names were often written in katakana.