Problem with sound intensity equation in displacement amplitude form

Question

The sound or wave intensity is defined by energy transfer rate with time (power) per unit of area:

$$ I = \frac{P}{A} \tag{1} $$

so this equation makes sense since the denominator is the area of the sphere and the numerator is the power of the sound or wave and it also explains why the further the source is, the lower the sound is:

$$ I = \frac{P}{4\pi r^2} \tag{2} $$

However, I don't understand this equation because all the quantities here are constant in an unchanged environment: wave speed, air density, angular frequency and displacement amplitude.

$$ I = \tfrac{1}{2} \rho v \omega^2 s_m^2 \tag{3} $$

So, according to this, the sound intensity is a constant and it doesn't express how intensity changes with the area that is affected by the distance from the source.

How to use it to calculate the intensity of a particular distance while the equation doesn't have distance quality?

Answer 1

Your last equation;

$$ I= \tfrac{1}{2} \rho v \omega^2 s_m^2 $$

is derived by taking the equation for a plane wave:

$$ s(x, t) = s_m \sin(\omega t - kx) $$

and calculating the kinetic energy associated with the wave. The calculation is described in this article from the Physics Hypertextbook.

For a spherical wave the prefactor $s_m$ is not a constant but decreases as $r^{-1}$. This means $s_m^2$ decreases as $r^{-2}$, and therefore both equations give an inverse square dependance of intensity on distance.