17.4:
Speed of Sound in Solids and Liquids
Solids and liquids are not easily compressible. When sound waves displace the constituent particles in these media, they display a tendency to restore to their original positions due to interparticle forces.
Recall that sound is a pressure wave. It travels through a medium as its constituents vibrate about their mean positions and transfer the energy to neighboring particles.
If the constituent particles are massive, it is harder to restore them to their original positions. Hence, the speed of sound through a medium is lower if its constituents are heavier.
Combining these concepts, the speed of sound in solids and liquids is given by the square root of the elastic modulus divided by the density of the respective medium.
17.4:
Speed of Sound in Solids and Liquids
Most solids and liquids are incompressible—their densities remain constant throughout. In the presence of an external force, the molecules tend to restore to their original positions, which is only possible because the constituents interact. The interactions help the constituents pass on information about external disturbances, like sound waves. Therefore, sound waves travel faster through these media. Compared to solids, the constituents in a liquid are less tightly bound. Thus, sound waves are relatively slower in liquids than in solids. Moreover, in solids, sound waves are not purely longitudinal; they also travel in the lateral direction.
The speed of sound in solids and liquids can be derived by applying Newton’s laws of motion on a column of the medium of propagation. Generally, the speed of sound in a solid or liquid is the square root of the ratio of the force restoring the particles to their equilibrium position, to the inertia resisting the restoration. The square root can be understood by recalling that the speed term appears as a square in the linear wave equation. The Young’s modulus determines the restoring force in solids, and its density determines the inertia. Hence, the speed of sound in a solid is the square root of its Young’s modulus divided by the density. Similarly, in liquids, the speed of sound is given by the square root of its Bulk modulus divided by its density.
Since density varies with temperature, the speed of sound in any solid or liquid medium implicitly varies with temperature.
This text is adapted from Openstax, University Physics Volume 1, Section 17.2: Speed of Sound.