I don't carry my bows on my bike. I stick them in the car when travelling with them. They are far too expensive to transport by bike. Both of us shoot compound BTW. We both used to shoot longbows but were lured to the dark side some years ago.
Thanks! I might just do that.........You can have a free try in May see here > http://www.archerygb.org/news/5274.php#.U07VDORH91Q and here
http://www.archerygb.org/bigweekend
Just find a club near you and pop down over that weekend.