# Is the sound speed and density relation different for compressible and incompressible media?

I'm studying "speed of sound in air" and I read that sound speed is inversely proportional to the density of the medium. However, I also know that sound travels fastest in solids, and that solids have highest density. If I'm not wrong (just my assumption), is the sound speed and density relation different for compressible and incompressible mediums? If I'm wrong, please tell me why and the correct answer to this question. Any help would be appreciated.

• Compression waves can't travel in an incompressible medium. The speed of sound is modeled as depending on both the medium density and the medium stiffness. Solids are generally much stiffer than gases. Mar 16 at 18:26
• I don't get what you mean by "stiffness of a medium". Are stiffness and density two different properties? And if they are, does sound speed relate with one and not the other in solids and gases?
Mar 16 at 19:03
• Part of learning physics is getting experience answering questions yourself using available online resources. These are questions you can easily answer on your own by reading the linked Wikipedia article. Mar 16 at 19:10
• The thing is, I don't trust Wikipedia that much because it's editable. Another problem is that whenever I make an assumption, I don't believe it as a fact without finding similar outcomes on the internet
Mar 16 at 21:10
• I think you'll find corroboration online that density isn't equivalent to stiffness, Wikipedia's fallibility notwithstanding. Mar 16 at 21:36

Conventional (i.e. compressive) sound is actually impossible in completely incompressible media, be they solids or liquids, and you can use the following generic relationship for the speed of sound in continua to demonstrate it:

$$c = \sqrt{\frac{\partial p}{\partial \rho}}$$

where $$c$$ is the speed of sound, $$p$$ is the pressure, and $$\rho$$ is the density.

Note that, in incompressible media, the density never changes—which means that any change in pressure, however big, cannot lead to a change in the density of the continuum. In a very heuristic rough sense, "the speed of sound in incompressible continua is infinite", in the sense that $$\frac{\partial \rho}{\partial p} = 0$$ and "therefore $$\frac{\partial \rho}{\partial p} = \infty$$". More accurately, it just means that perturbations in density for generic compressible media propagate faster and faster as the medium becomes more incompressible.

Often, compressibility is quantified directly through a property $$\beta$$ that is related to the change of pressure with density of a continuous medium:

$$\beta = \frac{1}{\rho}\frac{\partial \rho}{\partial p}$$

As a result, there is a direct connection between the compressibility of a medium and its speed of sound:

$$c = \sqrt{\frac{1}{\rho \beta}}$$

Therefore, the more compressible a material is, the smaller the speed of sound will be. And this tendency holds for any continuous medium.

• I'm sorry I didn't clarify my question. I just meant for a comparison between solid and gas