KrisBlueNZ
Sadly passed away in 2015
An H-bridge driver is a digital device and needs a square-wave input. If you feed a different waveshape into it, it will treat it as a squarewave anyway, so it makes no difference what waveshape you use. Only the duty cycle is significant, and you want that to be 50% so it would be best to use a squarewave.
Yes, definitely a sinewave would be best to drive the tweeter with. Any harmonics represent higher frequencies, which the tweeter can't reproduce and which are therefore converted into heat which could damage the tweeter.
An H-bridge driver with L-C filtering would be appropriate to use with a one-second burst of constant frequency. An H-bridge driver would be a good way to generate a large amount of electrical energy for the tweeter, but filtering would definitely be required.
Basically you would want to create a resonant (tuned) circuit (or L-C tank circuit) using the tweeter. This consists of an inductance and a capacitance in parallel. Since the tweeter is mostly inductive, you would need a capacitor in parallel with it, and some way to isolate the tank circuit from the driver while still transferring power from the driver to the tank circuit. This could be a resistor, an inductor, or a combination of both. Google TANK CIRCUIT for more information. I imagine that a lot of experimentation would be needed, and the resulting circuit and component values would be tied to a specific model of tweeter, but if it's worthwhile, it would give you a nice simple circuit that can deliver a large amount of energy to the tweeter with high efficiency.
I'll have to leave it to you to decide whether it's worth pursuing this idea. The other option is an audio amplifier that operates into a bridge-tied (BTL) load, or (less efficient and less powerful) a single-ended audio amplifier. Being linear (analogue) rather than digital, these can transfer a perfectly formed sinewave to the tweeter, so you would need a sinewave input.
Depending on your power supply capabilities and your output power requirements, a single-ended audio amplifier might be adequate. In this case you would not need a large value output capacitor - a few microfarads would be plenty, because you only need to reproduce one high frequency, rather than a range of frequencies including low frequencies as is the case with a generic audio amplifier.
Integrated (IC or Sanken-type module) power amplifiers are available with power ratings up to at least 60W, probably much more, and these are definitely the easiest way to go. The only reason you'd want to use discrete components is to save money if you're planning on mass production.
