Intensity and Sound Level — Question 4

Question 4

A level drop can determine an unknown geometric range

An isotropic point source is at height $h$ above a flat reference plane. One receiver is directly below it; another is $30.0\,\mathrm m$ horizontally away. The farther receiver measures $6.0206\,\mathrm{dB}$ less intensity level. Neglect reflections and absorption.

1. Infer $h$ using spherical spreading.

2. If the source power doubles, determine whether the inferred height or the level difference changes.

Question 4 - Solution

A level drop can determine an unknown geometric range

(a) Method and calculation

For equal source power, $I\propto1/r^2$. The level difference gives$$\frac{I_{\mathrm{near}}}{I_{\mathrm{far}}}=10^{6.0206/10}=4.000,\quad\frac{r_{\mathrm{far}}}{r_{\mathrm{near}}}=2.$$$$\sqrt{h^2+30^2}=2h\Rightarrow3h^2=900\Rightarrow h=17.32\,\mathrm m.$$

(b) Interpretation and further calculation

Doubling power multiplies both intensities by two, raising each level by $3.0103\,\mathrm{dB}$. Their ratio and level difference remain unchanged, so the inferred height remains $17.32\,\mathrm m$. The method eliminates unknown source power through the ratio.

Physical check

The farther slant distance is $34.64\,\mathrm m$. The result depends on isotropic spreading; directivity could imitate a range-dependent level change.