17.2:
Sound as Pressure Waves
Consider sound traveling through a medium. The longitudinal disturbances create a pressure difference between its successive columns, which then undergo oscillations.
Consider an undisturbed cylinder of the medium, with cross-sectional area A along the x-axis. Its longitudinal displacement, given by y, is a wave function.
As the wave travels, its ends at x-1 and x-2 are displaced by y-1 and y-2, respectively. If the latter is greater, the cylinder expands, and the pressure falls from the surrounding pressure.
Its initial volume is known, and its change in volume is derived. The fractional change in volume is then obtained.
Recall the definition of bulk modulus, from which the gauge pressure is obtained. On simplification, the gauge pressure is observed to be a wave.
The gauge pressure is maximum at points of zero displacement and minimum at points where the displacement is maximum.
Its amplitude is proportional to the displacement amplitude, the bulk modulus of the medium, and the wave number. So, it is inversely proportional to the wavelength.
17.2:
Sound as Pressure Waves
Sound waves, which are longitudinal waves, can be modeled as the displacement amplitude varying as a function of the spatial and temporal coordinates. As a column of the medium is displaced, its successive columns are also displaced. As the successive displacements differ relatively, a pressure difference with the surrounding pressure is created. The gauge pressure varies across the medium.
The pressure fluctuation depends on the difference in displacements between the successive points in the medium. A relationship between the instantaneous displacement of a particle in the medium and the gauge pressure can be obtained via the material's bulk modulus.
At points of compression, the pressure is the most positive because the medium's particles aggregate. At points of rarefaction, the particles are the farthest from each other, and the pressure is the most negative. In between, where the particles have maximum displacement, the pressure is zero.
Waves of shorter wavelengths have greater pressure amplitudes, and vice versa.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson: section 16.1; pages 510-512.
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/1-introduction: section 17.3; pages 860-861.