I am trying to figure out how to reach a very high level tone. However, after doing some googling, I am not sure how to do this without buying very expensive lab equipment. I don't need a lot of power, but I would like to be able to reach the sound level.
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Generating the electrical signal of 43KHz is trivial. And amplifying it up to speaker-level is pretty easy also. The more tricky part is getting drivers (speakers, tweeters) that will achieve the desired sound pressure level (SPL) over the area that you need.
If you were to reveal your budget, your target SPL, and the area/space/volume where you want to produce this sound, there are probably good suggestions you can get here.
The so-called air motion transformer tweeters will work pretty well up in that range. I have an old set of ESS/Heil AMT1 speakers, and the high end is extremely crisp (what I can still hear of it).
Here is a smaller tweeter of similar design that has response shown up to 40kHz:
It's 8 ohm, so most audio amplifier circuits that you can find in application notes will work with it, perhaps with minor adaptations. For generating a 43kHz sine wave, a function generator kit would work, or you could use a 555 and filter out some of the harmonics.