Intensity and Sound Level — Question 1

Question 1

Intensity uses RMS pressure rather than peak pressure

A progressive plane wave has $p=P\cos\omega t$ and $u=p/Z$, with $P=0.40\,\mathrm{Pa}$ and $Z=400\,\mathrm{Pa\,s/m}$. Instantaneous acoustic energy flux is $pu$. Define intensity level $\beta=10\log_{10}(I/I_0)$ with $I_0=10^{-12}\,\mathrm{W/m^2}$.

1. Derive the average intensity and calculate its level.

2. Calculate RMS pressure and show how using peak pressure in $I=p_{\mathrm{rms}}^2/Z$ biases the level.

Question 1 - Solution

Intensity uses RMS pressure rather than peak pressure

(a) Method and calculation

Averaging over complete cycles gives$$I=\langle pu\rangle=\frac{P^2}{Z}\langle\cos^2\omega t\rangle=\frac{P^2}{2Z}=2.00\times10^{-4}\,\mathrm{W/m^2}.$$$$\beta=10\log_{10}(2\times10^8)=83.010\,\mathrm{dB}.$$

(b) Interpretation and further calculation

The RMS pressure is$$p_{\mathrm{rms}}=P/\sqrt2=0.28284\,\mathrm{Pa}.$$Using $P$ as though it were RMS gives twice the true intensity and a level error $10\log_{10}2=3.010\,\mathrm{dB}$. The factor of two comes from cycle averaging, not from a different reference intensity.

Physical check

$pu$ has units $\mathrm{Pa\,m/s}=\mathrm{W/m^2}$. The impedance relation used here requires a progressive plane wave.