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.
