Water 氵 does the washing; 喿 — a tree full of chirping birds — lends the sound. 澡 originally meant washing the hands specifically; the whole body claimed the character later, and 洗澡 became the full-immersion phrase.
Northern China's 搓澡 scrub-down culture turns bathing into sport: public bathhouses employ professional scrubbers who exfoliate you like furniture being refinished. Yangzhou's motto: morning skin-wraps-water (tea), evening water-wraps-skin (bath).
Water 氵 and you chirping like happy birds 喿 — BATH time.
东北澡堂 Northeastern bathhouses are legendary social venues — deals are closed and friendships sealed between the sauna and the scrub table.