Is there a specific waveform that needs to be produced by the driver electronics in a solid state tesla coil?

Is there a specific waveform (for example sine, square, sawtooth, ramp, etc) that the driver electronics (an oscillator i think?) in a solid state tesla coil in order to properly drive the coil?

I'll preface this by stating that I have only a basic working knowledge of electronics; the majority of my opinions here are based on my knowledge of acoustics. However, there is a strong enough similarity between the physics of the two fields that I think I might be on the right track at least.

I'm guessing you would use a sine wave to drive it, since the oscillation in a normal Tesla coil is caused by tuned resonance between only two or three circuits.

Square and sawtooth waveforms are caused by complex addition of harmonics. Ideal (perfect) square waves have an infinite number of odd harmonics added fractionally to the fundamental; ideal sawtooth waves have infinite even harmonics.

Mathematically, a square wave is:

And a sawtooth wave is:

While it is possible to induce resonance between entities tuned to perfect harmonics, the exchange of energy wouldn't be nearly as efficient - not to mention that you would be tuning into a frequency that has only a fraction of the amplitude of the fundamental, as in the case of a square or sawtooth wave.

It looks like, after a brief search of Wikipedia, that the original design of the Tesla coil used a similar design to a Ruhmkorff coil. It is stated that this induction coil has an output that alternates positive and negative, but with one polarity being greater. I imagine this would be similar to a sine wave with some DC bias, as the induction of the wire would smooth out the AC from strictly "on-off" pulses (like a square wave) to a sine.

Further down the page, Wiki states that both coils must be perfectly tuned. This also implies to me that only a sine should be used.
You said exactly what I was going to say, but you said it much more clearly. And I didn't have the math or the research and I couldn't name the names or describe how the other wave forms were built etc. I was going to say that since a lot of the better tesla's that I've seen are run off of NST's that are plugged directly into the wall without any electronic control, a pure sine wave must be pretty good for a Tesla coil. But your answer was good too.