17.6:
Deriving the Speed of Sound in a Liquid
Consider a fluid in a lake, which is set into disturbance from the origin by a mechanical source at time zero in the x-direction and at a speed of v–x. Let the speed of the mechanical wave be v.
After a duration of t, the column of fluid that experiences the wave is contained within a distance v–t from the origin. If the fluid density and the cross-sectional area are known, the column's volume, its mass, and the momentum it has been imparted can also be determined.
The column's original volume decreases, and this is used to derive the gauge pressure change via the bulk modulus.
The impulse on the fluid column created by the disturbance is given by the gauge pressure change, the area, and the duration. The equation is then simplified.
By using the impulse-momentum theorem, the speed of the disturbance in the medium can be obtained. It depends only on the fluid's bulk modulus and its density.
17.6:
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.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson: section 16.2; pages 514-515.
- Verma, H.C. (2008). Concepts of Physics. Volume 1, Bharati Bhawan Pub. & Dis.; section 16.4; page 332.