An Etymological Dictionary of the German Language/wallen
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wallen (1.), vb., ‘to boil, bubble,’ from the the equiv MidHG. wallen, OHG. wallan, str. vb.; corresponding to OSax. wallan, AS. weallan, allied to OIc. vella, ‘to boil, bubble.’ From the same Aryan root wel (wol) is derived ModHG. Welle, ‘wave, billow.’
wallen (2.), vb., ‘to wander about, go on a pilgrimage,’ from MidHG. wallen, OHG. wallôn, ‘to wander, roam about, go on a pilgrimage’; corresponding to AS. weallian, ‘to wander.’ Allied to MidHG. wallœre, ‘traveller (on foot), pilgrim’; MidHG. wallevart, ModHG. Wallfahrt, f., ‘pilgrimage.’ The root wal (wallôn, rom wal-nô) is perhaps not different from the root of wallen (1). Comp. E. to walk (Teut. base wallaqôn) with AS. weallian.