3.10.2.2 - Sensitivity and Frequency Response

Intensity and spherical spreading

Intensity is the sound power transmitted per unit area perpendicular to propagation:

Here $P$ is the power crossing the specified area, not necessarily the total source power. A small source radiating acoustic power $P_s$ uniformly in all directions has, in a free field,

This assumes negligible absorption, reflection and obstruction. Electrical power delivered to a loudspeaker is not its acoustic output power unless the conversion efficiency is supplied. Doubling $r$ quadruples the spreading area and reduces intensity to one quarter.

For a uniform wave of intensity $2.0\times10^{-5}\,\mathrm{W\,m^{-2}}$ normally incident on an area $40\,\mathrm{mm^2}$, the incident power is $IA=2.0\times10^{-5}\times40\times10^{-6}=8.0\times10^{-10}\,\mathrm{W}$. This differs from calculating the total power of an isotropic source using a whole spherical area.

Hearing threshold and logarithmic response

The threshold of hearing at a specified frequency is the minimum intensity that a listener can detect. The conventional AQA reference is the threshold for good hearing at $1\,\mathrm{kHz}$:

This fixed reference is used at every frequency when calculating intensity level. It is not a claim that every listener's actual threshold is the same, or that threshold is independent of frequency.

The audible range spans many orders of magnitude: an intensity of order $1\,\mathrm{W\,m^{-2}}$, associated with painful sound, is $10^{12}$ times the reference. The discomfort threshold varies and is not a safe-exposure boundary.

A linear scale represents this range awkwardly. A logarithmic scale converts multiplicative intensity changes into additive level changes. Hearing is also non-linear: a tenfold intensity increase does not sound ten times as loud. At a fixed frequency a $10\,\mathrm{dB}$ increase is often perceived approximately as a doubling of loudness, but this is an empirical guide, not the definition of a decibel or an exact law for all sounds.

Decibels, ratios and combined sources

The intensity level $L$ compares sound intensity with a fixed reference using a base-10 logarithm:

The ratio inside the logarithm is dimensionless. $L$ is expressed in decibels (dB). A negative level is possible: it means $I<I_0$, not negative physical intensity.

For two sounds, subtracting their levels eliminates the reference:

An increase of $3.01\,\mathrm{dB}$ doubles intensity; $10\,\mathrm{dB}$ multiplies it by 10; $20\,\mathrm{dB}$ multiplies it by 100. A change from 60 to 80 dB is not an intensity ratio of $80/60$.

Combining spreading and level

A free-field source gives $80\,\mathrm{dB}$ at $3.0\,\mathrm{m}$. Its intensity is $10^{-12}10^8=10^{-4}\,\mathrm{W\,m^{-2}}$, and acoustic power is $4\pi(3.0)^2I=1.13\times10^{-2}\,\mathrm{W}$. At $6.0\,\mathrm{m}$, intensity is a quarter as large, so the level falls by $10\log_{10}4=6.02\,\mathrm{dB}$ to about $74\,\mathrm{dB}$.

Adding sources

For independent, incoherent sound sources, add intensities before taking logarithms. Two equal machines each contributing $5.0\times10^{-5}\,\mathrm{W\,m^{-2}}$ at the listener produce a total of $1.0\times10^{-4}\,\mathrm{W\,m^{-2}}$: the level changes from $76.99\,\mathrm{dB}$ to $80.00\,\mathrm{dB}$. Do not add 77 dB and 77 dB. Coherent tones require interference to be considered, so simple intensity addition is not universal.