Why do gases only transmit sound waves whose wavelength is long relative to the mean free path?
Since sound waves are longitudinal waves (generally in air) there needs to be some kind of interaction between neighboring "air" molecules (a mixture consisting mostly of $\ce{N2}$, $\ce{O2}$, $\ce{CO2}$ and $\ce{Ar}$). You can see that this is just not possible if the mean free path of the "air" molecules is larger than the desired wavelength. How would the "air" molecules interact on such distance if there are no "air" molecules to interact with? This interaction (i.e. sound of a particular wavelength) will only be possible if there are sufficiently many "air" molecules nearby to interact with, or formulated otherwise: if the mean free path is sufficiently much smaller than the desired wavelength of the sound. Of course, it is possible to make sound waves with a larger wavelength than the length of the mean free path, just not the converse.

One application I can think of lies more in Physics but whatever. In the days before Einsteins theory of special relativity was developed/accepted, there was a common consensus, which was supported by Maxwell himself, that light was travelling in a special reference frame called "The Aether". In this reference frame the speed of light equaled $c$. We now know of course that light can travel through vacuum and that there exists no such special reference frame as The Aether. There were various problems with this Aether theory, most famously the Michelson-Morley experiment that suggested that either the Aether did not exist or was moving for a weird reason with the exact same trajectory and speed as earth in space. Another problem, related to your question, was that the speed of waves in a medium is dependent on the particle density of that medium. And since (at least some) electromagnetic waves have an incredible small wavelength (the $\lambda$ of visible light for example was known), the density of this Aether medium had to be extremely high, yet it had to also weight almost nothing and be able to penetrate trough everything!

From this and other facts Einstein must have concluded that the Aether theory was in fact invalid. This idea ultimately lead to the theory of special and general relativity, both of which have many spectacular and far reaching consequences.