Measuring a sound is usually performed with a microphone of some sort - however this only measures one aspect of the sound wave. There are actually three distinct measurable characteristics of any sound:

• Power (Watts)
  Measures the energy output by a sound source, which is basically the sound's ability to do work.
• Pressure (Pa)
  Measures fluctuations about the local atmospheric pressure caused by the sound. The overall pressure is usually measured using a root-mean-square (rms) technique rather that peak-to-peak measures as the pressure fluctuates between positive and negative.
• Intensity (W/m²)
  The amount of sound energy passing within a specific cross-sectional area taken normal to the direction of propagation.

Unfortunately, objective measures of power, pressure or intensity in their raw units are usually not of too much use as they do not accurately reflect the response of the human ear - which is the main point of most sound measurements.

Human Response to Sound

Through extensive empirical testing it has been clearly shown that the ear's response to sound is proportionate, not to the absolute value of a stimulus, but to the ratio of the actual intensity of the sound to the threshold intensity. The threshold intensity is the very faintest sound that the ear can actually hear. Further to this, Fechner's law states that this relationship is a logarithmic one.

Sound level measurements are therefore generally referenced to a standard threshold of hearing, measured at 1000 Hz. For a standard human ear, this threshold can be stated in terms of sound intensity as:

or in terms of sound pressure as:

Whilst these are the minimum thresholds, the table below details the range of audible sounds that the ear can tolerate. The upper limit actually represents the threshold of pain - where the sound it so loud it actually hurts the ear and can cause physical damage.

As you can see from the table above, human pressure perception ranges from 20 micropascals up to 200 Pascal. This represents a considerable linear dynamic range (i.e.: 1E7 or 10,000,000 times). Because of this, and the way the ear works, it is convenient to firstly work with relative measurement scales rather than with absolute measurements, and secondly to logarithmically compress them.

Decibels

The units used to measure this ratio are called bels, in honour of Alexander Graham Bell. Two variables differ by one bel if one is ten (1E1) times greater than the other, and by two bels if one is one hundred (1E2) times greater than the other. The bel is still a very large unit so it is more convenient to divide it into 10 parts - hence the decibel.

The standard threshold values given above corresponds to exactly 0 (zero) decibels. The actual average threshold of hearing at 1000 Hz is more like about 4 decibels, but zero decibels is a convenient reference. Click here if you would like some basic mathematical information on logarithms as it is impossible to understand decibel ratios without them.

Loudness and Sound Levels

As opposed to a directly measurable characteristic, loudness is a subjective term describing the strength of the ear's perception of a sound. It is intimately related to sound intensity but can by no means be considered identical to intensity. The sound intensity must be factored by the ear's sensitivity to the particular frequencies contained in the sound. This is the kind of information contained in equal loudness contours for the human ear.

Figure 1 - Equal loudness contours showing the variation in sensitivity of the ear with both frequency and intensity.

It must also be considered that the ear's response to increasing sound intensity is a "power of ten" or logarithmic relationship. This is one of the motivations for using the decibel scale to measure sound intensity. A general 'rule of thumb' for loudness is that the power must be increased by about a factor of ten in order to sound twice as loud.

Thus, a direct physical measurement of intensity, or even sound pressure, is almost meaningless in sensory terms unless it is referenced back to a threshold value. As the decibel is a relative measure and is used to quantify both pressure and intensity levels, each measurement can be compared to a universally accepted standard threshold value to derive a Sound Level in decibels (dB). Sound levels are directly meaningful in sensory terms.

Sound Power refers to the absolute power of a sound source (in Watts) whereas Sound Power Level (SWL) refers to the magnitude of that power relative to a reference power (in dB). Thus:

where

W_ref = 1E-12 W/m²

You can use the following to experiment with this relationship (click in the other text box to update it when you make any change):

TEXTBOX = 10 log10(TEXTBOX / 1E-12 W/m²) dB

Similarly, Sound Intensity refers to the absolute intensity (in Wm-2) whereas Sound Intensity Level (SIL) refers to the magnitude of the sound intensity relative to the reference intensity. Thus:

where

Iref = 1E-12 W/m²

For a plane wave, the intensity (I) of a sound field is proportional to the mean-square of the pressure fluctuation (p²) - the actual relationship is given by: I = p²/(r_oc).

Therefore, as spherical waves approximate plane waves in the far field, the Sound Pressure Level (SPL) becomes:

or

where

P_ref = 2E-5 Pa

You can use the following to experiment with this relationship (click in the other text box to update it when you make any change):

TEXTBOX = 20 log10(TEXTBOX Pa / 2E-5) dB

Since pressure is far easier to physically measure than intensity, it is often useful to express SIL in terms of SPL. Since I = P²/(roc), by taking logs and substituting reference values:

Obviously, given the density component (r_oc), the last term is pressure and temperature dependant. At 20'C and 1 atm it calculates out to around 0.1dB (Given that roc = 410 rayls). Thus:

Acoustic Power

The concept of acoustic power is not readily considered by most people, due in part to a preoccupation with sound pressure, except of course loudspeaker designers. To put it simply, sound pressure is the result of sound power. Sound power is the cause, sound pressure is the effect. The aim in loudspeaker design is to maximise the amount of acoustic power output from the device whilst minimising input power and signal distortion. In many conventional loudspeakers, it can take as much as 100 electrical watts to produce one acoustical watt, yielding an efficiency of about 1%. Most cone-type transducers offer around 3% efficiency, with some advanced piezoelectric devices offering up to 20-25%.

Spatial Directivity

Once maximum acoustic power has been achieved, attention turns to its distribution. If all the power from a source can be focused over a small area, then a greater sound pressure can be achieved and, hence, a louder sound. With low frequencies it is not possible to direct them to any specific area as they tend to be omni-directional (see the barriers topic). However, high frequency sounds can be very directional, thus the shape of loud hailers, trumpets and horn speakers to guide the sound in a particular direction.

A device's directivity can be described with a term called "Q", which is basically an indication of its ability to confine the applied energy to a smaller unit area. Q, therefore, is a parameter of the speaker and its enclosure, not the electrical transducer inside it. A Q of 1 means that the source is essentially a point source, distributing acoustical energy spherically in all directions. A Q of 2 means a hemispherical distribution.

Thus, the same acoustic energy output over half the area will increase the the intensity, and therefore the pressure, by a factor of 2 in the area covered by the device. In an architectural environment this is quite useful as it allows more energy to be directed towards the audience and less to the walls and ceiling.

It is common practice to convert the Q rating of a device back into a decibel value known as the Directivity Index (DI). This is done using the formula:

Spectral Content

The power of different sources of sound can vary widely. Different sources can also produce sounds with varying spectral content. A very shrill gas leak may produce the same sound power as a lawnmower, but they will be two very different sounds. Later, we will see how to take this into consideration when determining sound levels.

Typical Sound Power Levels

From the IAC Noise Control Reference Handbook, 1989 Edition by Martin Hirschorn. © Copyright 1982, revised January 1989 by Industrial Acoustics Company.

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