Deriving the Speed of Sound in a Liquid

As with waves on a string, the speed of sound or a mechanical wave in a fluid depends on the fluid's elastic modulus and inertia. The two relevant physical quantities are the bulk modulus and the density of the material. Indeed, it turns out that the relationship between speed and the bulk modulus and density in fluids is the same as that between the speed and the Young's modulus and density in solids.

The speed of sound in fluids can be derived by considering a mechanical wave propagating longitudinally along a medium and by using the gauge pressure expression along with the impulse-momentum theorem.

The expression is valid for liquids as well as gases. In gases, an additional understanding of how the mechanical wave travels, along with the properties of gases, is used to derive an expression in terms of the temperature of the gas. The assumption here is that the propagation of sound waves happens so fast that it is adiabatic. This is because the compressed and rarefied gas elements do not have enough time to exchange heat with their surrounding elements.