I am working on a science project where I need to play a sine wave between 100 and 400 Hz for an extended period of time, say 10 minutes. I have read that pure tones can cause the speaker to overheat. What is a speaker that could meet the requirements of this load?
Sine waves aren't bad for drivers. Now if you run a sine wave that is a bit too powerful into the driver such that it's generating heat faster than it can sink the heat into the air, then it's going to burn up. But in general, if you buy a speaker that can handle what you want to do, and you put it in a reasonable enclosure, it won't burn up. I imagine what you read was probably advising you that music and movies will vary in level and they might send a loud signal to the speaker, but then they might quiet down for a bit, etc. whereas a test tone people generally just use one voltage and leave it. It's really only harmful if the input level is too high.
What speaker you need really depends on how loud you want to play and what size box you want to fit it in.
Note that a decent box is 100% required, or you would definitely blow pretty much any speaker you choose (unless you were to choose a very large speaker and then only drive it with a very small power).
The speaker would operate at high volumes. The louder the better. I had a 6 inch sub that I have already burnt up. I took it apart and the windings were totally fried. I do not want this to happen again.
Still could use more info (high volumes is somewhat more helpful than what you said before but still not all that specific). What kind of 6"? And what's your budget? What do you have for an amp?
If you just need a speaker I'd probably recommend you buy a box and a 10, 12 or 15 inch woofer or subwoofer. Woofer will do the 400Hz signal better than a sub and a sub would do a 100Hz signal better than most woofers. But a good enough subwoofer or woofer should both be able to cover you.
You could do a lot more output than any 6" driver relatively simply by going larger
(the box is a little bit smaller than ideal for the woofer, but it will still be close enough for it to give you 100Hz and 400Hz tones without issue)
I still don't know if this would be good enough, though. Compared to whatever you were using before it should be quite a bit better. But for all I know, you might still not be happy with it. Just to be clear, I did not include an amp in this as I'm not sure if you have an amp, or if you were using a powered 6" subwoofer or what. (Subs can be passive or active and I don't know which sub you had.)
You could also look at pre-built subs like Dayton or BIC that are known to be good for the money and might be good enough for your purpose. However some powered subwoofers come with amplifiers that will run a low-pass filter and some don't let you disable it. If you buy a sub with a built-in amp, make sure the lowpass filter can be disabled.