Speaker cone shape

Essentially the old Infinity. They had some good ... and some bad. I am a fan of midranges in this size. Not many really good. 

Well this has been interesting.  So though there is science involved and calculations to make, and I dare say the implementation of aspects of others previous design successes to assist with the design of a good speaker system, the bottom line is once you have done all that and built your speaker, suck it and see seems to be the final process.

I’m not sure what you mean by “suck it and see”. Your post was about cone shape. Good speaker design is full of challenges. But cone shape isn’t one of them. Buy what sounds best to you. 

What I meant by "suck it and see", is reading the posts, despite any science or past design experiments, be that of cone shape, cabinet or any other aspect of the speaker design, the bottom line seems to be you have no guarantee that it will sound any better to you than something put together in your garage out of available parts. You design it, build it, take all the available measurements, but it's not until you listen to it, do you find out how good or bad it is. Just like a new recipe, despite having all the best ingredients you don't know until you taste it. Add to that the fact that tastes in how speakers sound vary, along with all the other variables, and what you said becomes the deciding factor be they $$$ latest tech or not, if they sound good to you, they are good.

to the OP,
Let's go back to understand what physically is happening to the cone. The voice coil attached to a cone will move the cone when the voice coil moves forward and backward. The entire cone does not respond all at once. Instead, a compression wave and a bending wave is launched into the cone by the voice coil. The compression wave causes no sound. The bending wave travels down the cone and induces a compression wave in the air.

As the bending wave travels to the far edge of the cone (at the surround), the air compression wave travels away from the cone surface. Due to the difference of the wave speed in the cone and the wave speed in the air, the effective wavefront at the plane of the cone surround can vary from a concave shape to a convex shape, with a plane wave between the two extremes.

Since the speed of the wave in the cone is typically slower than in the air, the cone angle can be designed to produce a convex spherical wavefront from the cone, propagating out  of the speaker into the forward half-space.

In the specific case of the Walsh speaker design with the apex of the cone pointed upward, the cone radiates sound from the convex side of the diaphragm. Since the sonic velocity in the cone much higher than in the air, a cylindrical wavefront is propagated in a 360-degree horizontal direction. With appropriate control of sound speed variation in the diaphragm, a waveform can be made to be spherical, radiating like a point source over a wideband of frequencies, without crossovers.

The reason most electromagnetic speakers use multiple drivers is that the designer wants each cone to move a if it were a rigid surface, which it's not. So the cone is restricted in frequency range where it acts as if it were moving all at once. For higher frequencies, a smaller driver is used to act as if it were moving all at once, because the wavelength in the cone (and in the air) is much shorter than for lower frequencies. And, so forth, which means you need crossovers to direct the right frequency range to each driver.