Acoustic Point Source

• Let $\mathbf{x}=(x,y,z)$ denote the Cartesian coordinates of a point in 3D space

• Point source at $\mathbf{x}=\mathbf{x}_1=(x_1,y_1,z_1)$

• Listening point at $\mathbf{x}=\mathbf{x}_2=(x_2,y_2,z_2)$

• Propagation distance:
  $$r_{12} = \left\Vert\,\mathbf{x}_2 - \mathbf{x}_1\,\right\Vert = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$

Acoustic pressure peak amplitude (or rms level) at $\mathbf{x}=\mathbf{x}_2$ is given by

$$p(\mathbf{x}_2) = \frac{p_1}{r_{12}}$$

where $p_1$ is the peak amplitude (or rms level) at $r_{12}=\left\Vert\,\mathbf{x}_2 - \mathbf{x}_1\,\right\Vert=1$

Notice that pressure decreases as $1/r$ away from the point source