# Does the speed at which sound travel depend on the volume (amplitude) of the sound?

Lets say you have a plank is you hit it once and get t time if you hit is 2x as hard will it travel t/2? will it be the same or will it travel only slightly faster?

• It will travel at almost exactly the same velocity. Nonlinear effects will affect speed of sound very slightly - but like with all waves, the velocity of propagation is independent of amplitude when linear approximations are valid. – Floris Sep 12 '14 at 3:59

A plank is a complicated example to choose because it's a composite material with a complicated structure. A better choice would be a piece of iron or some other homogeneous material.

In that case the speed of sound is given by:

$$v = \sqrt{\frac{K + \tfrac{4}{3}G}{\rho}}$$

where $K$ is the bulk modulus and $G$ is the shear modulus. The bulk modulus is how easy the material is to compress, and the shear modulus is how easy it is to deform sideways. Under most circumstances these moduli are constants, so the speed of sound is a constant. So when you hit the metal twice as hard the speed of sound is unchanged. In other words the speed of sound is not affected by the intensity of the sound.

However the moduli are only approximately constant for relatively small deformations, and for large deformations their values will change. If you can make the sound so intense that the deformation of the metal enters the non-linear region then the moduli will change and therefore so will the speed of sound.

Another unrelated effect is that the compression caused by the sound will heat the metal, and the moduli generally decrease with temperature. So if the sound is so intense that it causes local heating the moduli will decrease and so will the speed of sound.

However I should emphasise that you need very high energies for the moduito change appreciably. Under most circumstances you can consider the speed of the sound to be constant and not affected by the volume.

• Than what makes something intense? lets say you have a string of atoms and you hit it. then you hit it twice as hard. what will the difference be between the two? – Ray Kay Sep 5 '14 at 15:53
• @RayKay Amplitude of the sound wave. – Ajedi32 Sep 5 '14 at 16:10
• @RayKay The difference will be the amplitude by which the atoms displace from their equilibrium positions. In most every day cases, this displacement is quite small and so the restoring force felt by the atom is close to linearly proportional to the displacement (and in the linear case, speed of sound is constant). If the atom is hit hard enough, it will move far enough from its equilibrium position that non-linear effects will become significant enough that the linear approximation no longer applies (and, in general, the speed of sound is not constant in the non-linear case). – Kyle Sep 5 '14 at 17:38

I guess you mean to ask - is the amplitude of the vibration proportional to the speed of the sound waves it produces?

The speed of sound in an ideal gas for relatively small amplitudes ($\frac{\Delta P}{P} \ll 1$) is $v=\sqrt{\frac{\gamma P}{\rho}}$ where $\gamma$ is the adiabatic constant (i.e. $PV^\gamma=const$), P is the average pressure, and $\rho$ is the density of the gas.

In your normal day life, this approximation is very good, and the speed would not change in a meaningful manner as a function of the amplitude of the sound wave.

But if you produce a very loud sound (much louder than anything you have ever heard, which would cause your eardrums to burst and would probably bring about more adverse effects) - than the speed of this sound wave would be different (but would still not follow your original formula).

Volume is not strict word for describing sound (look, how many meanings in acoustics it has) http://en.wikipedia.org/wiki/Volume_%28disambiguation%29

"Loud" sounds are basically those of big amplitude. And amplitude of wave and its speed are two different things that have not much in common. Hitting a plank harder will make louder sound (the plank will vibrate with bigger amplitude, generating "louder" sound", however this will be also sound of different frequency).

As you can see here, speed of sound depends only on elastic modulus and density, however in some cases the amplitude can affect the speed of sound (Sound could compress air so much, it would change its density).