"Ordinary bows" ... the longbow was the standard in England anyway, the bows used in France were pretty similar, that and during the time crossbow mercenaries were being bought and brought over from Genoa. I'd also like to point you to this battle:
http://en.wikipedia.org/wiki/Battle_of_Verneuil ... the battle of Verneuil where Lombardian plated Italian mercenaries ride right the through the English line after being pelted with arrows ... why? Due to the previous video:
http://www.youtube.com/watch?v=D3997HZuWjk
Now archers are useful for slowing down the enemy and weakening them before they get to your lines, NOT as a way to combat armour, there are plenty more useful and better weapons to do that, the volume of arrows and projectiles in the air and hitting your opponent is what makes the bow so formidable ... not what it can and can't penetrate ... because if those cavalry at Verneuil had come back instead of going to a baggage train ... the English army would of been wiped out without a scratch to the french.
Please note Agincourt only took place before this battle by a mear 10 years ... thats not much time at all. Please also note with Agincourt that the french made it past the line of English Archers right into the baggage train ... much like Verneuil ... the English archer and bow is not as powerful as you think it is, its a weapon just like any other. With agincourt the mud posed no problem to the rider of the horse ... as demonstrated here:
http://www.youtube.com/watch?v=WMuNXWFPewg , the problem was with the horse, it slows down a bit, and if the mud is that bad then obviously infantry would have the same problem as slipping and sliding around, the English probably also had this problem but had the advantage of being static, which means they didn't need to move anyway.
Agincourt is always bigged up as a major victory ... but in just over 10 years the french counter attack again at Verneuil ... Agincourt was not really a major victory as historians think because otherwise the French would still be ruled by the English, it was a victory out of many losses, and after Henry V everything went downhill.