First off, its important to remember that such models are just models. Most of the wave propagation equations you are using assume a linear transfer medium... aka the atmosphere.
When you get into extreme situations like your infinite frequency, those assumptions start to break down entirely. They cease to be good models of how the world works. As such, you need to re-define the problem in terms which are better captured.
Take the sonic boom as an example. We assume "pressure information cannot propagate faster than the speed of sound in the medium." We also assume that all objects have a "boundary" layer of gas particles with a relative velocity of zero with respect to the object. Put them together, along with an object traveling faster than mach 1, and something has to give. You can't achieve all of those goals simultaneously. What ends up happening? We get a "shock wave" that's a few nanometers thick where we can't use our nice easy wave-mechanics models of the atmosphere. We have to treat the molecules in the air as individual particles in all their glory, and all sorts of funny effects come up. Everything behaves reasonably on both sides of this boundary, but they behave so poorly in those few nanometers that we typically just treat it as a discontinuous boundary and solve the equations on both sides.
So, recognizing that the models will almost certainly break down, what happens in your infinite frequency case? The first question is how do we define the wave you want to look at? We don't have enough information. Frequency is not enough. You also need amplitude or energy if you want to talk about what something will sound like.
So what if you hold amplitude constant and increase the frequency to infinity? Well, the answer is that you tear the universe apart. The power of a wave is proportional to both the square of the amplitude and the square of the frequency: $P\propto (Af)^2$. If you just increase the frequency towards infintiy while holding A constant, your power increases unbounded. If this wave is any larger than a single point, that means unbounded energy densities. You start to get into relatavistic effects and end up either swallowing up the universe in a black hole or otherwise ripping apart space-time as we know it! Let's not do that.
So what happens if you hold energy constant, to avoid this relatavistic nightmare? Well, to do this while frequency goes up, amplitude must go down. At infinite frequency, the amplitude must be zero, so the particles in the medium are not being moved any meaningful distance.
This will clearly violate many of the assumptions of the wave equation. It's going to be hard to argue that the molecules bashing together at a huge-but-finite rate are going to meaningfully propagate an infinite frequency wave. Realistically speaking, gases will exhibit a high frequency roll-off in this respect. They will attenuate waves as they get to higher frequencies.
As a result, your infinite frequency wave, with some reasonable assumptions to fill in the missing pieces, will sound like nothing at all. It will not transmit through the air meaningfully, because the medium attenuates such frequencies.
And even if it did get the sound to your ear, your ear also attenuates high frequencies. It has a bandpass like behavior built into it that prevents us from hearing sounds higher than roughly 22kHz.