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# SOUND LEVELS, (Power, Intensity, Pressure) # What is sound

Sound can generally by defined as vibrating waves in elastic media. These waves transfer energy without permanently transferring mass. Elastic media are gases, vapors, liquids and solids.
The human ear is capable of hearing vibrations in air in the frequency range of 20 to 20000 Hz. Underneath 20 Hz one speaks of intra-sound, above 20000 Hz of ultra-sound.

A human ear actually perceives the effective sound pressure level. This is a scalar that does not contain directional information. Direction estimation is done by the brain by evaluation of the time difference between the signals of both ears. The time difference occurs when a longitudinal sound wave passes the two ears under an angle.

So both scalars and vectors play a role in the technical representation of sound.

# Types of waves

Waves progress in elastic media. The type of wave depends on the medium. In acoustics the most important types are the longitudinal wave and the bending wave.
Sound in air is a longitudinal wave. The face of the wave is normally flat, but can be spherical if it is caused by a point source. Bending waves are important for structure born sound. Bending waves progress in walls and plates. These bring the air in motion, causing radiation of sound. Other types of waves are dilatation waves and transverse waves.
An attractive visualisation of longitudinal and transverse waves can be found at The University of Messina

The speed of a longitudinal wave can be calculated: in which:
c = speed of sound in m/s
p = average pressure in Pa = density in kg/m3 = specific heat ratio

For bending waves the formula is: with:
E = modulus of elasticity in Pa
d = thickness of the plate in m = surface mass of the plate in kg/m2

f = frequency in Hz

The formula shows a wave speed that is a function of the frequency. High frequency waves will travel faster than low frequency waves (dispersion). This can be observed at railways, when a train is coming the high frequency waves will travel faster and arrive first.

# Sound power, intensity and pressure

Progressing sound can be represented by the sum of different sinus shaped waves. In an open field sound is stronger near the source and weaker further away as the energy of the source is distributed over a larger area. Both the sound intensity level and associated sound pressure level will be lower further away.

Underneath it is explained how these can be quantified.

## Sound Power

A noise source radiates a certain amount of energy per unit of time. If one draws an area where the surface is perpendicular to direction of the energy flow and there are no losses of energy between the source and the surface, conservation of energy leads to: or, if a surface that was not perpendicular was chosen: In a lot of cases the intensity can be taken as evenly distributed over the surface. Many technical calculations are made like this, for example a sphere around a point source. The symbols:
P = Sound Power in Watts
I = Sound Intensity in W/m2
S = Surface in m2

## Sound Intensity

The intensity of sound is by definition the average power that is transmitted in the direction of progression, so intensity is a vector: it possesses magnitude and direction. For a longitudinal wave with a flat wave front the relationship between sound pressure and intensity can be proven to be: with: = effective sound pressure

The effective sound pressure is the root mean squared value (RMS) of the signal: # Sound Levels and dB

The human ear can perceive sound pressure over a very large range. The threshold of hearing at a frequency of 1000 Hz is peff = 2.10-5 Pa. This threshold of hearing can be reproduced in a laboratory quite well and was chosen as the reference value.
Above peff = 100 Pa is painful.

The figures for Sound Power can range from 10-9 W, for a whispering voice to 106 W for a jet engine with afterburner.

These figures are not very handy, so the logaritmic scale was introduced. Using a logaritmic scale requires reference values. These reference values are (ISO 1683-2):
Pref = 10-12 W
Iref = 10-12 W/m2
pref = 2.10-5 Pa (=N/m2)

Using these the definition of the sound levels are (all in dB):
Sound Power Level: Sound Intensity Level: Sound Pressure Level: These values for these acoustic parameters were taken with care to obtain simple relationships for atmosperical air. The numerical value of sound pressure level and sound intensity level are the same as: The last term has the value 0.13 for atmospherical air and thus can be neglected.
This makes SPL = LI.

## Relationship Sound Power level and Sound Pressure Level

So the relationship between sound power level, sound pressure level and the surface is: 