Question

What is the speed of the sound in a perfectly rigid rod?

Answer 1

Bulk modulus:

1. Bulk modulus is the constant that describes the elastic properties of an object (solid or liquid) when it is under pressure on all surfaces.
2. The applied pressure reduces the volume of a material, which returns to its original volume when the pressure is removed.
3. The mathematical expression of the bulk modulus is $B=\frac{\Delta P}{{\displaystyle \raisebox{1ex}{$\Delta V$}\!\left/ \!\raisebox{-1ex}{$V$}\right.}}$, where $B$ is the bulk modulus, $\Delta P$ is the change in pressure, $V$ is the initial volume, $\Delta V$ is the change in volume.

An illustrative diagram of the bulk modulus is shown in the figure below.

Explanation:

1. For a perfectly rigid rod, the bulk modulus of the material can be $B=\infty$.
2. So the speed of sound will be $v=\sqrt{\frac{B}{\rho }}=\infty$, where $v$ is the velocity of sound, $B$ is the bulk of modulus, $\rho$ is the density.
3. Hence, the speed of sound in the perfectly rigid rod is infinite.