# Why is it that sound at standard conditions always travels the same speed?

Say you have a small fire cracker. When it explodes, it imparts some kinetic energy to the surrounding air molecules, and they propagate that kinetic energy outward through collisions.

Intuitively, I feel that if you have a bigger fire cracker that imparts more kinetic energy, then the air molecules should move faster and collide faster with their neighbors, so the speed of sound would be faster in that case.

Of course, this isn't correct - the speed of sound is pretty much constant for a given density and pressure.

Could someone help me build a better mental picture of what's happening?

The first air molecules to experience the fire cracker blast are moving faster, but that kinetic energy is quickly uniformly dispersed to neighboring molecules through many collisions. So the kinetic energy is dispersed but there is still a preferential (radial) direction of the energy

"Sound" is due to something moving slower than, well, the speed of sound. "Shock waves" travel faster than that.

Really Big explosions can create shock waves that move faster than sound for a long way, but your firecracker examples are just the usual form of sound-based noise.

The sound of the explosion is propagated by the motion of the atoms due to their thermal energy. One way to see this: Far from the firecracker, the energy of the explosion has not increased the energy of the air molecules at all: their mechanism for transporting sound energy isn't about the firecracker at all.

To put it another way: The energy of the firecracker is tiny compared to the energy of the air molecules. A cubic meter of room-temperature air is about 1 kg of molecules, each with thermal velocity of 460 m/s. That's about 100kJ of energy. The fire cracker is much, much less than that, and the sound is spread over much, much more than that.

• I don't think this helps me understand why, regardless the energy of the firecracker (as long as it's below a certain threshold where a shock wave would be created), the sound wave propagates at the same speed -- the speed of sound. I want to have an understanding of the mechanism whereby different initial energies result in waves of the same speed. – EternusVia Sep 22 at 21:07
• Sorry, there's really not more to say. Do you understand the basic mechanism of sound propagation? – Bob Jacobsen Sep 22 at 21:20
• I think the idea that the energy input from the fire cracker is much smaller than the energy of the surrounding air is what helped me understand. – EternusVia Sep 22 at 21:46

The speed of sound $$v$$ in a material is given by: $$v = \sqrt{\frac{B}{\rho}},$$ where $$B$$ is the bulk modulus (quantifying how easily compressible the medium is) and $$\rho$$ the density of the medium. This formula applies for "normal" propagation away from the source. In this region, we can assume $$\rho$$ and $$B$$ to be constant.

Sound is an oscillation of the air (or whatever medium) molecules, i.e. a sequence of rarefactions and compressions:

The amplitude of the oscillation is what gives sound volume: a large amplitude means a great difference in pressure between the compressed and rarefied region, i.e. a big force on the eardrum, which causes pain and possible rupture.

This amplitude is dictated by the initial energy of the event creating the sound wave, in this case the explosion of the firecracker. But it does not necessarily change the speed of sound itself.

Think of an harmonic oscillator:

The period (frequency) of oscillation is always the same. A higher kinetic energy of the particle just causes a larger amplitude of the oscillation, i.e. the ball has a higher turning point. But it takes the same time to cover an entire period.

So a more powerful firecraker results in a larger amplitude of the oscillation, i.e. a louder sound and a stronger pressure differential between compressions and rarefactions. Away from the source, however, the speed of this oscillation is the always the same. As for the period of a harmonic oscillator.

### Close to the source: shock waves

I said earlier that away from the source you can take the density $$\rho$$ of the medium constant. I.e. the source does not change the characteristics of the medium itself.

Close to the source of the explosion, you cannot. The abrupt release of energy in a limited space can dramatically alter the characteristics of the medium, such as its density.

The shock waves cause a rapid change in the density of air $$\rho$$, which result in a change of refractive index, like in here: