Is it valid to wipe the head with the same water remaining on the hands after washing the arms, without renewing it, and does not renewing the water invalidate the ablution?

Wiping the head with the wetness of the hands after washing them is not sufficient for ablution according to the Shafi'is, Hanbalis, and Hanafis. Rather, new water must be taken to wipe the head because the remaining wetness is water that has been used to remove ritual impurity. Therefore, the ablution and prayer are considered invalid and must be repeated. As for the Malikis, ablution is valid if the wetness of the hands is sufficient to wipe the head, though it is disliked. It is sunnah to repeat the prayer within its prescribed time, but repeating it afterwards is not obligatory.