# The effect of frequency and amplitude on the speed of sound

I don't do physics, but a sound-related question came up in a project. I've googled extensively but I still can't get a hang of the basics. The following are two related questions.

Assume the source vibrates at a frequency $f_1$ and amplitude $A_1$ and produces a longitudinal wave with a wavelength of $\lambda_1$.

Question #1:

Now let's assume that we change the amplitude of vibration to the greater value of $A_2$, but keep the same frequency $f_1$. Now according to theory, the speed of sound will not change. However, doesn't vibrating surface cover greater distance in the same amount of time, compared to when it oscillates with lower amplitude $A_1$? Which, I would assume, should mean that the speed of wave propagation is higher?

Question #2:

Now let's assume that we keep the same amplitude $A_1$, but change the frequency of the source to the greater value of $f_2$.

Similarly, vibrating surface covers greater distance in the same amount of time compared to when it oscillates with at a lower frequency $f_1$. Is that correct? Similarly, shouldn't the speed of wave propagation be faster?

• "The speed of sound in an ideal gas depends only on its temperature and composition. The speed has a weak dependence on frequency and pressure in ordinary air, deviating slightly from ideal behavior." - en.wikipedia.org/wiki/Speed_of_sound Dec 4, 2017 at 2:25

Answer #1: The amplitude of an acoustic wave is proportional to the compression of the particles of the propagation media. Waves with $A_1$ and $A_2$ have no difference in propagation speed, $A_2$ is simply more intensely compressed than $A_1$.

The most helpful equation for the speed of sound in your case is: $$c = \sqrt{\frac{\partial p}{\partial \rho}}$$

If equilibrium pressure, $p$, and the medium density, $\rho$ are constant there will be no change in acoustic velocity.

Answer #2: This is slightly more complex because your answer depends on the medium. Some media (especially solids like metal bars or sheets of ice on lakes) are dispersive which means that changing $f$ changes acoustic velocity.

There's a pretty good description of what dispersion means on this site:

The nature of solid materials in a rest state is such that the chemical bonds in their susbstance prefer specific distances between the atoms or molecules. Stretching a material causes it to want to snap back. Compressing it makes it want to uncompress. These are properties of chemical bonds, which are complex interactions between shared electrons and nuclear spacings.

Note it's still a property of the medium that causes dispersion to occur.

Going back to the equation above it's easy to see that in most media there's nothing about the frequency of sound that causes it's acoustic velocity to be faster or slower. In the special case of a dispersive medium, it is a property of the form of the medium that causes the changes in acoustic velocity.