Is there an intensity estimation algorithm working solely in time domain

Given a second of sampled sound pressure signal, speed of sound and medium density, can the average sound intensity be estimated without performing DFT, that is in linear time by simply traversing all samples?

Could you provide\link a proof that it is correctly evaluating\approximating this definition from wikipedia:
$I_{avg}&space;=&space;\frac{1}{T}\int_{0}^{T}p(t)\mathbf{v}(t)dt$

A lot of examples deal with pure tonal sine waves, where particle velocity member can be transformed to pressure, and amplitude is clearly defined, but I'm struggling to find an example for random signal.

Particle velocity and sound pressure are related by the field impedance. The field impedance is a function of the media and the boundary conditions of the field.

For a reasonably shaped wave in a reasonably well behaved medium, that impedance is simply $$\rho \cdot c$$, i.e. density of the medium times speed of sound. For air that is about 400 Pa/(m/s). There is no phase shift and not a lot of frequency dependency.

In this case the intensity signal can be calculated as

$$I(t) = \frac{p^2(t)}{\rho \cdot c}$$

This will only get you the magnitude. Keep in mind that intensity is a vector with the same direction as the particle velocity. If you want to integrate Intensity to get actual power you need to take into account the angle between the Intensity vector and the normal vector of your integration surface.

If you only have the pressure signal, you have to make some assumptions around the direction of the vector. Pressure is a scalar and there is no spatial information contained.

• And "reasonable" wave becomes unreasonable when I choose the sampling rate too little, or when the sound is a shock wave, I presume? Nov 30, 2018 at 9:26
• @Boris-Barboris: "reasonable" always depends on your specific requirement, but in general it means: "in the far field, away from the source". "locally flat, can be approximated by a plane wave over short stretches" "in an amplitude range where the medium is mostly linear". It has nothing to do with the sample rate, this is just pure physics. Dec 1, 2018 at 10:17