What do "elastic property" and "inertial property" mean in this passage about the speed of sound in elastic media?

Question

I was reading the chapter Sound Waves from my textbook and I came across this line that I am unable to comprehend properly.

Here's how it goes -

The speed of an mechanical, transverse or longitudinal, depends on both an inertial property of the medium (to store kinetic energy) and an elastic property of the medium (to store potential energy). Thus, we can generalise $v = \sqrt{\tau\over{\mu}}$ which gives the speed of a transverse wave along a stretched string, by writing $v = \sqrt{\tau\over{\mu}} = \sqrt{\frac{ \text{elastic property}}{\text{inertial property}}}$, where (for transverse wave) $\tau$ is the tension in the string and $\mu$ is the string's linear density. If the medium is air and the wave is longitudinal, we can guess that the inertial property, corresponding to $\mu$, is the volume density $\rho$ of air. What shall we put for the elastic property?

I want to know the meaning of inertial property, elastic property, how that relates to the formula given for $v$ and the answer to the question in the very last line of the paragraph.

Answer 1

If you think of the classic simple harmonic oscillator that consists of a mass $m$ connected to a spring.

The spring is said to have a spring constant $k$ that means if the mass is displaced from its equilibrium position by a distance $x$, then the spring will try and get the mass back to its equilibrium position with a force $- k x $. The negative sign means the force is a restoring force, so if you compress the spring then the spring will push the mass away, while if you elongate the spring, it brings the mass back in.

Mathematically you can write $$ m \frac{d^2 x}{d t^2} + k x = 0, $$ which can be written in the form $$\frac{d^2 x}{d t^2} + \omega^2 x = 0.$$

In the above you have $\omega = \sqrt{ k / m} $.

In your problem, the inertial property is akin to the mass $m$, while the elastic property is akin to the restoring force.

Answer 2

The "inertial property" refers to anything that stores energy as kinetic energy, like a chunk of mass. The "elastic property" refers to anything that stores energy as potential energy, like a squeezed spring. In the mechanical engineering discipline of dynamic systems modeling, the generalized inertial property is called inertance and the generalized elastic property is called compliance.