cavalry fought with lances and swords; bows, if used, were few and far between; IIRC, the lakhmids were more likely to use them, and even then. the lance was about the size of a Xyston, made of canes from the murraan tree (citing Al-Jahiz). swords were either the short, native arabian swords, or, should the owner be of good luck, the long persian ones (3ft).
now, how did they do it? It helped that their leaders knew what they were doing. In particular, the exploits of Khalid Ibn Al-waleed are of interest; he never really attacked the enemy cavalry head on, but when they were either scattered, or from the rear (citing the accounts of Yarmouk). and he only used them at the last possible moment, for reasons which I will elaborate. It also healped that the enemy cavalry had been decimated by the recent romano-persian wars, especially the sassanids (evidently the byzantines weren't totally exhausted; they had a well accoutered army in northern Syria; what they didn't have was coordination, as yarmouk clearly shows.)
besides, cavalry was actually unimportant to the Arabs in the early days, as there was a dearth of cavalry. most of the troops were infantry. so the more accurate question is: how did arab infantry deafeat the cataphracts?. lots of factors were involved:
1-the romans and persians were exhausted, particularly the persians. both were still able to crush the arabs who came out (battle of the bridge is a good example), but seemed to have lacked the willpower or motivation
2-there was better coordination among the arabs than either enemy-any account of walajah, dhat-as-salaasil, or yarmouk show so. the adversaries were also not as imaginative for some reason.
3-Khalid ibn al-waleed happened to be an Arab-Muslim
4-the Byzantines made the stupid mistake of not listening to jabalah ibn-al'aiham, prior to yarmouk.
5-the persians made the even dumber mistake of underestimating their foe, if one is to believe the accounts.(figured they learned better at Callinicum-the ghassanids were spectacular there. too bad Belisarius ran away without good reason.
6-and most importantly: the Arabs evidently wanted to win more than either adversary did.
so in short: its not the armor you have or the weapons you use, but how you use them, that decided this one.
EDIT: oh, and in almost every major battle, the Arabs were outnumbered, especially in the cavalry department.
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