Why is sound intensity proportional to the square of sound pressure not to sound pressure alone?

Question

I am trying to understand the physical principles behind the sound intensity and sound pressure. As far as i know, sound intensity is proportionate to the squared sound pressure. Can someone explain me in simple words why the pressure needs to be squared?

Answer 1

I'm guessing you want an intuitive feel for why $I \propto P^2$ rather than a rigorous derivation, so this answer is somewhat hand waving in nature. Anyhow, sound pressure is obviously just pressure i.e. force per unit area, but sound intensity is energy flow per unit area and the energy flow is proportional to $P^2$.

To see why this is take a plane wave and consider some planar element of area $A$ in the oscillating medium. The work done to move this element a distance $dx$ is force times distance, and force is pressure times area, so the work done is:

$$ dW = PAdx $$

If this happens in a time $dt$ the power, i.e. work per unit time, is:

$$ \frac{dW}{dt} = P A \frac{dx}{dt} = P A v $$

where $v$ is the velocity at which our planar element is moving. Divide by the area, $A$, to get the power flow per unit area, i.e. the intensity, and we get:

$$ I(t) = P(t) v(t) $$

Note that these are the instantaneous values of pressure and velocity. The velocity $v$ is not the velocity of the wave, it's the instantaneous velocity of the moving bit of the medium. Likewise $P$ is the instantaneous difference of the pressure from ambient pressure at our moving element.

Now this is the bit I don't have a quick proof for: for a plane wave the instantaneous velocity is proportional to the pressure (both oscillate in phase):

$$ v(t) = kP(t) $$

for some constant $k$, and therefore substituting in the equation for $I$ gives:

$$ I(t) = kP(t)^2 $$

This gives the instantaneous sound intensity and we normally want the average, or more precisely root mean square value. For this just use the RMS value of the sound pressure:

$$ I = kP_{RMS}^2 $$

Answer 2

As sound pressure is inversely proportional to distance r from the source, and Intensity is inversely proportional to r^2, hence sound intensity is proportional to squared sound pressure, you can check distance law of an acoustic source.