Intensity and Sound Level — Question 7

Question 7

Equivalent level is an energy average over time

A signal has intensity level $80\,\mathrm{dB}$ for $10\,\mathrm s$ and $60\,\mathrm{dB}$ for $50\,\mathrm s$. Define the equivalent level by $I_{\mathrm{eq}}=\sum I_jt_j/\sum t_j$, using the same reference for all levels.

1. Find the equivalent level over the full minute.

2. Find the fraction of total acoustic energy per area contributed by the ten-second interval. Compare with the arithmetic mean of the two levels.

Question 7 - Solution

Equivalent level is an energy average over time

(a) Method and calculation

Normalize by the reference intensity before averaging:$$\frac{I_{\mathrm{eq}}}{I_0}=\frac{10(10^8)+50(10^6)}{60}=1.75\times10^7.$$$$\beta_{\mathrm{eq}}=10\log_{10}(1.75\times10^7)=72.430\,\mathrm{dB}.$$

(b) Interpretation and further calculation

The high-level interval contributes$$\frac{10(10^8)}{10(10^8)+50(10^6)}=\frac{20}{21}=95.24\%.$$The arithmetic mean $70\,\mathrm{dB}$ ignores unequal durations and the logarithmic scale. Even the time-weighted decibel average, $63.33\,\mathrm{dB}$, is not an energy average.

Physical check

The equivalent level lies between $60$ and $80\,\mathrm{dB}$. This is an acoustic energy calculation, not a hearing-risk assessment.