Intensity and Sound Level — Question 6

Question 6

Absorption can be isolated after removing geometric spreading

A spherical wave has intensity $I(r)=C r^{-2}e^{-\alpha r}$, where $\alpha$ is an intensity attenuation coefficient. Levels measured at $r_1=10\,\mathrm m$ and $r_2=20\,\mathrm m$ differ by $8.00\,\mathrm{dB}$, with the farther level lower.

1. Infer $\alpha$.

2. Find the additional distance that halves intensity by absorption alone. State the corresponding pressure-amplitude attenuation coefficient.

Question 6 - Solution

Absorption can be isolated after removing geometric spreading

(a) Method and calculation

Take the logarithm of the intensity ratio:$$\beta_1-\beta_2=20\log_{10}(r_2/r_1)+\frac{10\alpha(r_2-r_1)}{\ln10}.$$$$\alpha=\frac{[8.00-20\log_{10}2]\ln10}{10(10)}=0.045577\,\mathrm{m^{-1}}.$$Geometric spreading accounts for $6.0206\,\mathrm{dB}$; the remainder is absorption.

(b) Interpretation and further calculation

The absorption-only half-intensity length satisfies$$e^{-\alpha\ell_{1/2}}=\tfrac12,\qquad\ell_{1/2}=\frac{\ln2}{\alpha}=15.21\,\mathrm m.$$Because intensity is proportional to pressure amplitude squared, pressure carries the factor $e^{-\alpha r/2}$. Its attenuation coefficient is $\alpha/2$, in addition to the geometrical $1/r$ factor.

Physical check

A measured drop below $6.0206\,\mathrm{dB}$ would imply negative $\alpha$ and contradict passive absorption under this model.