# How do we get supersonic bullets?

I recently answered a question on the WorldBuilding forum about grenades and bullets. One of the things that came up was that I argued smokeless powder in a rifle round could detonate, but was challenged on that. Commenters said that smokeless powder only deflagrates during normal use.

This, however, leaves me with a question. How can we accelerate a bullet to supersonic speeds using only a sonic speed pressure wave? As the bullet approaches the speed of sound, shouldn't the pressure wave be pushing the bullet less effectively? It strikes me that a bullet traveling at the speed of sound should not be able to be pushed by a pressure wave at the speed of sound.

How does this work?

• Black powder can send a bullet supersonic, too. My understanding is that it deflagrates. – Don Branson Feb 4 '19 at 19:45
• Deflagration (Lat: de + flagrare, "to burn down") is subsonic combustion propagating through heat transfer; hot burning material heats the next layer of cold material and ignites it. Most "fires" found in daily life, from flames to explosions such as that of Black powder, are deflagrations. This differs from detonation, which propagates supersonically through shock waves, decomposing a substance extremely quickly. wikipedia. – Captain Giraffe Feb 5 '19 at 2:10

## 2 Answers

The speed of sound increases with increasing pressure. Assuming ideal behaviour the relationship is:

$$v = \sqrt{\gamma\frac{P}{\rho}}$$

or equivalently:

$$v = \sqrt{\frac{\gamma RT}{M}}$$

where $$M$$ is the molar mass.

In a gun barrel just after the charge has gone off the gas is under very high pressure and very hot, so the speed of sound is much higher than under ambient conditions.

Deflagration means that the combustion moves through the fuel slower than the speed of sound in the fuel. It doesn't say anything about the speed of the resulting gas, or how it compares to the speed of sound in that gas (and the speed of sound in solids is generally higher than that of gasses). The gas that is released from the combustion isn't at equilibrium, so properties such as "pressure", "temperature", and thus "speed of sound" aren't fully defined for it.

• This does limit the combustion process though. If the medium is moving faster than the flame front, unburned fuel downstream of the flame front isn't ever burned. This is a huge oversimplification. There is a lot of nuance to this speed relative to what etc, but the upshot is that unconfined there is a limit on the speed of the resulting gas dependent on among other factors the speed of combustion. – drjpizzle Feb 6 '19 at 14:47