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I'd suggest doing some kind of fund raising. Talk to your band director/boosters about it. The boosters may also give you some cash.

With the director's and booster's permission, have something like a car wash, or sell something, or some sort of fund raising that will go directly to buying new tubas.

In the marching band I am in, I used to have an awful, beat-up old fiberglass Yamaha sousaphone that I had patched up in several places with duct tape to stop the leaking and buzzing (you might try this as a temporary fix!).
What I did to raise money for a new horn was I went around door-to-door in my neighborhood with this tuba asking for donations and playing a tune or two for everyone I met. Between the shock of finding me honking away at the door and the sight of this wreck of a tuba, I convinced many people to throw in a few bucks for me.
While this may not raise enough for a new tuba quickly, it is a good way to start and to raise community awareness of your plight. The four tuba players could go around together, maybe even learn a tuba quartet! Don't be shy, and have some fun. Good luck!

Your best bet is to work the system first. If the system fails, go all out to surpass/bypass/end-run/outwit it.

As someone who has experience as an educator, I've dealt with folks who have a real problem if they're bypassed. More often than not, it's because the bypass has highlighted their inactivity or ineptitude.

If your band teacher has an education worth a cent, he should know that the sound of an ensemble is built from the bass-end up, and not from the drum section. If he doesn't understand this, it highlights, once again, his ineptitude; he may not understand anything -but- drums. New instruments make it easier for you to sound better, which will help the band sound better, which will make him look better. He might understand that.

Not knowing the full situation, I would wonder the age of the previous drum equipment; once a drum shell starts falling apart, there's little to do that will keep it in a usable condition. To most people, an old tuba can be duct-taped/wired/glued for a while. In all fairness, the drums, in his mind, were a higher priority. Has any communication with him shed light on whether any plans are currently in the works for new/newer instruments?

Are used instruments a possibility? In some school districts, used equipment may _not_ be purchased; this, of course, adds to the cost, which can then be prohibitive.

Make sure you've communicated fully with your band director. Unfortunately, I've come across a number of colleagues who cannot deal with someone going out on their own, door-to-door, looking for donations. A backlash could ensue. I encourage you to work with him to put a concrete plan in place. Get all the PARENTS of the people in your section involved. (Unfortunately, I've also known many who don't seem to take anything seriously until the parents get involved.) Do you have a band parents organization of any sort? They can be a powerful fundraising machine.

As far as the horns go, bracing with duct tape is neither dependable nor effective for more than the first five minutes. Go to your local hardware store. In the automotive section you will find hose clamps of many sizes. In the plumbing section, find clear plastic hose that will accomodate the hose clamps. You can use these to wrap around and catch the loose braces most anywhere on the horn. For that matter, it may also be the way to silence the other buzzes. Part of the problem is that convertible tubas are poor; they're made to work well either way, and, as a result, don't work well either way.

That's my two cents.

Best of luck to you.