I work at one of NY's major airports. You see those pvc tube cases quite a bit whether it's for fishing rods or bows. You won't be able to carry it aboard so you'll give it up at check in & pick it up on arrival. Just make sure you put a TSA openable lock on the unscrewable end. When it's inspected, snap it shut. If the TSA has to inspect it again somewhere along the line they have the master keys to open it.
I think mine cost $8. to make w/o the lock. They're about $6. ea.
Also, if you're bringing an arrow case, tape it to the bow tube . .it'll go as one piece! They charge extra per piece for additional baggage.
Larry