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.