17.9:
Intensity and Pressure of Sound Waves
Consider a sound wave propagating in the x-direction and the displacement of the particles denoted by the wave function y. Recall the expression for a pressure wave.
The particle velocity is given by the time derivative of the displacement, which is obtained.
Recall the expression for power as a product of force and velocity. The power per unit area is given by the product of the force per unit area—which is the pressure—and the velocity. The intensity is the time average of this quantity.
Substituting the particle velocity and pressure expressions and taking the average over time, the intensity is derived.
It is further simplified by writing the wave number in terms of the wave velocity, which is related to the bulk modulus.
Recall the relationship between the pressure and displacement amplitudes. Using these three equations, the intensity can be expressed in terms of the pressure amplitude, the density of the medium, and the wave velocity.
17.9:
Intensity and Pressure of Sound Waves
The intensity of sound waves can be related to displacement and pressure amplitudes by using their wave expressions and the definition of intensity. The critical step to achieve this is to write the power delivered by the particles on the wave as the product of force and velocity and simplify the force per unit area as the pressure. The velocity of the medium's particles can be derived from the displacement.
Unlike the time average of a sinusoidal term, which is zero since it is positive and negative for the same amount of time every cycle, the time average of the square of a sinusoidal term is 1/2.
The sound intensity expression in terms of the displacement amplitude is also related to the wave's frequency. This expression explains why high-frequency waves create the same intensity, even for small amplitude vibrations, compared to low-frequency waves.
The sound intensity expressed via the pressure amplitude does not depend on the wave frequency, making it easier to discuss the intensity's relationship with the medium's properties.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson: section 16.3; pages 518-519.
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/1-introduction: section 17.3; page 861.