drow·sy / ˈdrouzē/ • adj. (-si·er, -si·est) sleepy and lethargic; half asleep: the wine had made her drowsy. ∎  causing sleepiness: the drowsy heat of the meadows. ∎  (esp. of a place) very peaceful and quiet: a drowsy suburb.DERIVATIVES: drow·si·ly / -zəlē/ adv.drow·si·ness n.

drowsy XVI. prob. based on the stem of OE. drūsian be languid or sluggish, f. *drūs-, var. of base drēosan fall (cf. DREARY). See -Y 1.
Hence, by back-formation, drowse be inactive, or heavy or dull with sleep. XVI.