Why does maximum sound absorption occur when angular frequency equals the reciprocal of relaxation time?

In the study of relaxation phenomena, sound propagation through a gas acts as a periodic disturbance, creating a sinusoidal variation in pressure and temperature. The system's ability to maintain chemical equilibrium depends on the relationship between the frequency of these oscillations and the system's inherent relaxation time ($\tau$).

When the frequency is very low, the system remains in equilibrium with the sound wave. When the frequency is very high, the system cannot respond at all, and the state remains constant. Maximum sound absorption occurs at intermediate frequencies where the system's response lags behind the pressure variations. Specifically, this peak in energy dissipation occurs when the angular frequency ($\omega$) of the sound wave matches the reciprocal of the relaxation time, expressed as:

$$\omega =\frac{1}{\tau }$$

At this condition, the phase lag between the external stress and the system's response is optimized, leading to the greatest conversion of acoustic energy into other forms, such as heat, through the chemical reaction or energy redistribution processes.12