The Collaborative International Dictionary of English v.0.48:

Drowse \Drowse\ (drouz), v. i. [imp. & p. p. Drowsed (drouzd);
   p. pr. & vb. n. Drowsing.] [AS. dr[=u]sian, dr[=u]san, to
   sink, become slow or inactive; cf. OD. droosen to be sleepy,
   fall asleep, LG. dr[=u]sen, druusken, to slumber, fall down
   with a noise; prob, akin to AS. dre['o]san to fall. See
   Dreary.]
   To sleep imperfectly or unsoundly; to slumber; to be heavy
   with sleepiness; to doze. "He drowsed upon his couch."
   --South.
   [1913 Webster]

         In the pool drowsed the cattle up to their knees.
                                                  --Lowell.
   [1913 Webster]