Source : Webster's Revised Unabridged Dictionary (1913)

Drowse \Drowse\, v. i. [imp. & p. p. {Drowsed}; p. pr. & vb. n.
   {Drowsing}.] [AS. dr?sian, dr?san, to sink, become slow or
   inactive; cf. OD. droosen to be sleepy, fall asleep, LG.
   dr?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.

         In the pool drowsed the cattle up to their knees.
                                                  --Lowell.

Source : WordNet®

drowsing
     adj : half asleep; "made drowsy by the long ride"; "it seemed a
           pity to disturb the drowsing (or dozing) professor"; "a
           tired dozy child"; "the nodding (or napping)
           grandmother in her rocking chair" [syn: {drowsy}, {drowsing(a)},
            {dozy}]