the perception of sound – part 1

In order to understand the acoustics of any installation, it’s important to understand how sound is perceived and calculated.

A sound source oscillates and causes small pressure fluctuations in the surrounding air, gas or fluid, causing particles to start moving outwards. With the mass and compressibility of the air, the fluctuations are transmitted to the listener’s ear.

The pressure fluctuations are referred to as sound pressure p. The sound pressure is superimposed on the static atmospheric pressure p0 which depends on time and space. The sound source radiates a spatially distributed sound field with different instantaneous sound pressures at each moment. The observed sound incident at any point has two main distinguishing attributes – timbre and loudness.

The physical quantity for loudness (or amplitude) is sound pressure p, measured in N/m2.  The physical quantity for timbre is frequency
f, measured in Hertz (Hz) or cycles per second, 1Hz = 1/s.

Our human hearing range starts at about 16Hz and ranges up to 16000Hz or 16kHz. The ultrasound is above that frequency range and the infrasound below it, both of them being of technical interest, too.

A sound incident that can be described by a sine curve  in the time-domain is  called a pure or harmonic tone. A harmonic tone can only rarely be observed in natural sound conditions.

Relationship between absolute sound pressure and sound pressure level
Sound pressure
p(N/m², eff.)
Sound pressure level
L (dB)
Situation
2 x 10-50hearing threshold
2 x 10-420forest, slow winds
2 x 10-340library
2 x 10-260office
2 x 10-180busy street
2 x 100100pneumatic hammer, siren
2 x 101120jet plane during take-off
2 x 102140threshold of pain, hearing loss

A sound incident that can be described by a sine curve in the time-domain is  called a pure or harmonic tone. A harmonic tone can only rarely be observed in natural sound conditions.

The physical quantity for timbre is frequency f, measured in Hertz (Hz) or cycles per second, 1Hz = 1/s.

Our human hearing range starts at about 16Hz and ranges up to 16000Hz or 16kHz. The ultrasound is above that frequency range and the infrasound below it, both of them being of technical interest, too.

A sound incident that can be described by a  sine  curve  in the time-domain is  called  a pure or harmonic tone. A harmonic tone can only rarely be observed in natural sound conditions.

Even the sound of a musical instrument contains a superposition of several harmonic tones typical of the instrument. However, an arbitrary sound can be represented as a  sum of harmonics, their frequencies and their amplitudes.

The frequency components – the acoustic spectrum – of a sound can be extracted through spectral analysis similar to the spectral analysis of light.

In terms of sound pitch, the human ear perceives the tonal difference of two pairs of tones fa1, fa2 and fb1, fb2 equally, if the ratio – not the difference – of the two pairs is equal, that is, if we have fa1/fa2 = fb1/fb2.

So, for example, we perceive the transition from 100Hz to 125Hz and the transition from 1000Hz to 1250Hz as an equal change in pitch. This relative tonal impression is  reflected  in the subdivision of the scale into octaves – a doubling of frequency – and other intervals such as second, third, fourth and fifth, which is commonly used in music.

 Weber-Fechner Law

It is not only this tonal impression that a stimulus R has to be increased by a certain  percentage to be perceived as an equal change in perception. It is true for other human senses as well.

In mathematical terms, the increment of a perception ΔE is proportional to the ratio of the absolute increase of the stimulus ΔR and the stimulus R, so ΔE = kΔR/R where k is a proportionality constant.

Moving towards infinitesimally small variations dE and dR,  integration  yields E=2.3 k  lg  (R/R0) where lg  is  the logarithm to the base 10 and R0 the threshold stimulus, at which the perception starts. This relation is known as the Weber-Fechner Law.

The perception of loudness also follows the logarithmic Weber-Fechner Law, since the human ear is faced with the task of perceiving very quiet sounds, such as falling leaves in a quiet countryside, as well as very loud sounds, such as the roaring of a nearby waterfall.

Indeed, humans are able to perceive sound pressures from 20×10−6N/m2 to approximately 200N/m2 where the upper limit is the human pain threshold.

Human hearing covers about seven orders of magnitude of loudness, which is an exceedingly large physical interval. It is therefore handy to use a logarithmic measure when quantifying sound pressure technically, instead of the physical sound pressure itself.

The sound pressure level L is defined as:  L=20 lg (p/p0)=10 lg (p/p0)2 where p0=20×10−6N/m2.

The reference value p0 corresponds to the hearing threshold at a frequency of 1kHz as the hearing threshold depends on the frequency.

The specification decibel dB is not  a unit but indicates the use of the logarithm to the base 10. The factor 20 is chosen in a way that corresponds to our perception – if two sounds differ by 1dB, we just perceive a difference in loudness.

Assigning sound pressure levels maps the range of sound pressure covering seven powers of ten to a scale from 0 to 140dB.

It is remarkable that even the sound pressure of 200N/m2 related to the sound pressure level at the pain threshold is still only a small fraction (1/500) of the static atmospheric pressure of about 105N/m2.

Often, several sound pressure levels  have to be summarised to one. Signals originating from sources independent from each other, such as different technical devices or machines  or different speakers, are called incoherent signals. It is nearly always justified to assume incoherent signals.

It can be shown that the relevant squared root-mean-square  value  (rms-value)   of the total signal is the sum of the individual squared rms-values. That is, the squared rms- value of N incoherent signals is given by p2eff    =∑n p2i=1 eff,i       .