Ok. I'm a longbow shooter myself and it just so happens I work for a major airline as a mechanic. I've flown with mine multiple times. I personally still get nervous about it. I also would NOT use standard PVC. I'd use the heavier schedule 40 type. I've had my tube busted once though the bow was fine luckily. The airline paid to replace the tube.
The issue is generally two things... But there will be several points I'll try to make
First off, the obvious one is the length of the piece does not lend itself to traveling well on conveyor systems that nearly always have turns in them. If your item gets jammed in a bend on the conveyor, the belt does not stop, other suitcases continue to come along and everything backs up. Think of a log getting stuck in a culvert pipe during a storm and other debris washing in behind it. It builds up until somebody stops the belt or something gives. Because the conveyors are typically covered, it might be a case of nobody realizes there's a blockage until it blows the top off and suitcases start falling on the ground. Because conveyors nearly always run from 3-4 stories up in the terminal down to the ramp level outside, it can be a long way down. The general rule in most airports is that items more than a certain length get put on a special conveyor with no bends OR they get hand carried down to the baggage room. People travel with skis and golf clubs pretty often. They should know how to handle oversize items. The simple truth is, it doesn't always happen. Think new guys working and honest mistakes made. At baggage check in, be sure you ASK about the hand carry or separate baggage belt for bulky items. Marking the tube with "Fragile, Do Not Bend". Or "Hand Carry if Needed Please" may be a good idea. Nothing brings anxiety on a hunt more than getting to your destination and being the last guy standing at the baggage carousel because your bow case didn't come up yet... In many airports there is a totally separate belt for such items or, if not, they may get carried up to the baggage claim office by hand. That office may or may not be near the carousel where your regular bags came up. A good gate agent should know or be able to find out what you need to do to get your special item at your destination point. Asking ahead of time can save a lot of worry. Remember, always be polite at check in. Don't be an a-hole about your bow case or it's value. You can certainly express your opinion or concerns but do it nicely and you should get polite, informative responses in return.
Second point, PVC tubes are round and they ROLL easily. Remember those conveyors? Not ALL of them have side guards. Also, the belt loaders that get pulled up to the side of the plane... They have side guards but many times are not put up or only put up on one side. A REALLY good idea would be to connect two tubes together or attach something to the tube so it can't roll. I've seen where guys used a metal banding tool to attach triangular wood blocks for this reason. Baggage carts are another place to worry about. They have side doors made of a heavy tarp like material and are supposed to be closed when moving them. Sometimes they are broken, sometimes the driver just forgets. I once watched my two gun rifle case come sliding out onto the ramp as the driver made a sweeping turn to position the string of carts to be unloaded onto the plane. Luckily, it didn't come open!
Another issue with ANY large item is if there is an overweight situation. Too many bags, too much weight etc and they will start pulling suitcases to send on a later flight. Guess which cases they pull first? The biggest, heaviest ones they find. Granted, a bow in a plastic tube is not heavy but it's bulky. Just something to think about. Honestly, I don't think that's really an issue these days but years ago before baggage restrictions it was not uncommon. Back in the 80's, people didn't want to use carry on's because it was a hassle Today, it is preferred and checked bags are limited.
Anyway, that's been my experience. Others may vary. I know some people going on hunts prefer to ship their gun or bow ahead of time if possible. This let's you get word that everything has arrived safely even before you leave home. It puts a burden on whoever is running the hunt
though.
Oh, and because tubes CAN be broken, usually on the ends, I always make sure my bow is inside a nice fleece case and there is a pretty good size piece of foam in each end. I also cut a pair of foam "donuts" that I slide over each limb to about the mid point and which need to be compressed a bit to fit inside the tube. This keeps things from rattling around in the tube and more importantly, if an end cap gets broken, it keeps the bow from sliding out. It saved mine when the cap was broken.