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The Inverse-Square Law & Sound Power and Sound Pressure

The law of decreasing power per unit area (intensity) of a wave front with increasing distance from the source is known as the inverse-square law, because intensity drops in proportion to the inverse square of the distance from the source. Why is this? It is because the sound power from a point source is spread over the surface area of a sphere ( S ), which from elementary math is given by:




where r is the distance from the source or the radius of the sphere, as shown in the diagram.


If the original power of the source is W watts, then the intensity, or power per unit area ( I ) at distance r , is:



For example, if the power of a source was 0.1 watt, the intensity at 4m distance would be:




The sound intensity level (SIL) of this signal in decibels can be calculated by comparing it with the accepted reference level of 10^-12 Wm-2 :





The sound power which had passed through 1m 2 of space at distance r from the source will pass through 4m 2 at distance 2 r , and thus will have one quarter of the intensity.


The amount of acoustical power generated by real sound sources is surprisingly small, compared with the number of watts of electrical power involved in lighting a light bulb, for example. An acoustical source radiating 20 watts would produce a sound pressure level close to the threshold of pain if a listener was close to the source.


The amount of heat produced by the dissipation of acoustic energy is relatively insignificant –the chances of increasing the temperature of a room by shouting are slight, at least in the physical sense.


Sound pressure is the effect of sound power on its surroundings. To use a central heating analogy, sound power is analogous to the heat energy generated by a radiator into a room, whilst sound pressure is analogous to the temperature of the air in the room. The temperature is what a person entering the room would feel, but the heat-generating radiator is the source of power. Sound pressure level (SPL) is measured in newtons per square meter (Nm-2). A convenient reference level is set for sound pressure and intensity measurements, this being referred to as 0 dB. This level of 0dB is approximately equivalent to the threshold of hearing (the quietest sound perceivable by an average person) at a frequency of 1 kHz, and corresponds to an SPL of 2x10-5 Nm-2, which in turn is equivalent to an intensity of approximately 10x12 Wm-2 in the free field.


Sound pressure levels are often quoted in dB (e.g. SPL=63 dB means that the SPL is 63 dB above 2x10-5 Nm-2). The SPL in dB may not accurately represent the loudness of a sound, and thus a subjective unit of loudness has been derived from research data, called the phon.


Bibliography:


  1. Sound and Recording, Sixth Edition, Francis Rumsey and Tim McCormick.

  2. Designing Sound, MIT

  3. Sound Design, Maurizio Giri.

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