They do correlate to each other, though; barrel volume to frequency, duration to 'overbore'/excess case volume and also gas velocity (which in turn is related to pressure),
Absolutely true. I only aimed to point out that there is no consistent general equation for dB
SI/kPa
pressure at the muzzle is arguably correlated to powder volume as well (supposedly all powders generate roughly the same volume of gas, from what I've read).
Yes, it would be. And yes, smokeless powders, as a rule of thumb, expand to 14,000 times their volume as a solid. I would point out, though, that you have to account for barrel length. All else being equal, a barrel that continues to house the bullet for a longer period of time is going to result in the pressure attenuating further before the projectile exits the muzzle. This is one of the basic principles behind suppressors as well; contain, vector and attenuate the blast to keep a significant portion of it from exiting.
Powder burn rate can, of course, affect this, most notably with handguns; if the powder does not burn completely before the bullet exits, you may have unburned powder exiting the muzzle. If it never burns, it never expands to gas, hence less total pressure. But that is really inconsequential and academic minutia.
Frequency should be the same, seeing as same bore, same barrel length (for 16" carbine). Pressure is markedly higher in the AR in all lengths; I honestly wonder if the 5.7 isn't getting close to ambient after 16". Duration favors the AR as well, owing to it's larger case volume.
Honestly don't know enough about acoustics to speculate about the frequency aspect. I would surmise, though, that myriad factors come into play there, including barrel harmonics, air density, temperature, and so on.
I also don't know that the volume directly correlates with the duration, though I'm inclined to agree with your theory.
There is one factor that goes unmentioned, though, and that is the overpressure bubble. When a super-sonic high pressure round is fired, a spherical bubble of super-sonic gas is forced from the muzzle,
That's not exclusive to super sonic rounds; All smokeless powders expand at a super sonic rate. The component that
is unique to super sonic rounds is the sonic crack the bullet produces.
The boundary of the bubble is a pressure discontinuity or shockwave, with higher pressure air on one side and ambient on the other, with no gradient between them. It is literally like having your ears boxed by a palm if it is large enough to engulf your person; there is bulk force involved due to the pressure differential. Even a super short 223 can generate enough velocity to get the gas bubble formed at the muzzle, and the size of the bubble is directly proportionate to the mass of expanding gas inside the bubble (or pressure & temperature, if you will). The more powder generating the bubble, the farther it can expand before the pressure inside is the same as ambient, and the pressure wave coalesces into a mere sound wave. At the same time, the shorter the barrel, the closer you start to the bubble in the first place.
I concur with all but the highlighted portion. Pressure does not make sound, and sound is not a pressure wave. Sound is vibration, which is a
change in pressure at a frequency. Air at 1 bar (atmosphere @ sea level) is silent, as is air at 10, 20, 100 atmospheres. Which is why your charged compressor in the garage makes no noise (unless it's leaking). But create changes in pressure, you get noise, such as that obnoxious pulsating when your car window is at just the wrong height going down the road.
But again, pressure and sound do not have a fixed relationship that is easily quantified. Smacking a piece of 1/4" sheet steel with a ball peen hammer is far more painful to the ears than the thud of a bass drum, but creates far less pressure. Conversely, that pulsating pressure created by the partially open car window is painful to your ears, but at a dB level that would be completely tolerable absent the high pressure component.
The science of acoustics is very interesting, and admittedly one that I need to spend a whole lot more time on. Far more complex than most people ever care to discover, for sure. To most, noise is either pleasant or it's not, and they don't really care about the why.