An Etymological Dictionary of the German Language/Annotated/wallen
wallen (1.), verb, ‘to boil, bubble,’ from the the equiv Middle High German wallen, Old High German wallan, strong verb; corresponding to Old Saxon wallan, Anglo-Saxon weallan, allied to Old Icelandic vella, ‘to boil, bubble.’ From the same Aryan root wel (wol) is derived Modern High German Welle, ‘wave, billow.’
wallen (2.), verb, ‘to wander about, go on a pilgrimage,’ from Middle High German wallen, Old High German wallôn, ‘to wander, roam about, go on a pilgrimage’; corresponding to Anglo-Saxon weallian, ‘to wander.’ Allied to Middle High German wallœre, ‘traveller (on foot), pilgrim’; Middle High German wallevart, Modern High German Wallfahrt, feminine, ‘pilgrimage.’ The root wal (wallôn, rom wal-nô) is perhaps not different from the root of wallen (1). Compare English to walk (Teutonic base wallaqôn) with Anglo-Saxon weallian.