Appendix A

Intensity and Loudness Measures of Sound

The intensity I of a sound is a physical measure of its acoustical energy flow to the ear, measured in W/m². Intensity is proportional to the square of the distance to the sound source assuming anechoic conditions. A logarithmic scale is used to represent the extremely broad range of intensities to which the ear is sensitive. The sound intensity level IL in dB is the log ratio of a given intensity I to a reference I₀ of 10^-12 W/m², the lower threshold of hearing at 1 kHz.

(A.1)

The physical quantity of sound pressure level (SPL) often is used since average pressure variations are easier to measure than intensities (and thus sound intensity levels). SPL also is measured on a logarithmic scale relative to a reference D p₀ of 2 x 10^-5 N/m².

(A.2)

For a traveling wave, SPL and IL are equivalent because the minimum threshold I₀ corresponds to the average pressure variation of D p₀, and because intensity is proportional to the square of D p. For standing waves, SPL must be used because I and therefore IL cannot be measured [12].

While IL and SPL are not functions of the frequency of the sound wave, the psychoacoustic magnitude of loudness does depend on frequency. In general, the ear is most sensitive around 3 kHz and less sensitive at much lower and higher frequencies. The loudness level LL of a pure tone is given by the SPL of another tone at 1000 Hz that is judged to be equally loud. LL, measured in phons, is therefore a measure of equal or constant loudness. A tone whose LL is twice that of another is not twice as loud. The "subjective loudness" L, measured in sones, must be used for such relative loudness measurements. Increasing the loudness level LL by 10 phons roughly doubles the subjective loudness L. Note that LL still is considered a physical magnitude while L is a psychoacoustic magnitude.

The relationship between L and the intensity or average pressure variation of a single sound source is approximated as in Eq. (A.3) [12].

(A.3)

Here C₁ and C₂ are parameters that are functions of frequency.

When a sound source produces components at different frequencies or when more than one sound source exists, the calculation of L is more complicated and takes on one of three forms [12]. If the frequencies of the different components all fall within the so-called critical band of the center frequency, L is related to the sum of the intensities of the individual components as:

(A.4)

An engineer may wish to fit Eq. (A.4) into a typical rms-type equation, but perceptual phenomena are not constrained by standard mathematical models.

"When the frequency spread of the multitone stimulus exceeds the critical band, the resulting subjective loudness is greater than that obtained by simple intensity summation …, increasing with increasing frequency difference and tending toward a value that is given by the sum of individual loudness contributions from adjacent critical bands [12]." Eq. (A.5) describes L in the limit as the spread between the component frequencies increases. Note that this does not take into account masking effects if the individual loudnesses of the components differ greatly from each other.

(A.5)

While it seems like Eqs. (A.4) and (A.5) may cover all situations (for components inside and outside the critical band), things become more complicated when frequency differences between individual components are large. In this case, people tend to focus on the loudest or highest pitch component and assign the sensation of total loudness to that single component. Eq. (A.6) explains this last situation.

(A.6)

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Jim West, University of Miami, Copyright 1998