Sound Pressure Maths

Sound Pressure

When people talk of sound level, they usually refer to it in decibels.  This may be when thinking about OSHA requirements, how loud a rock band is, jet engines, or town ordinances. So what are decibels?  To answer that, let us first look at sound pressure.   Sound is measured by changes in air pressure.  The louder a sound is, the larger the change in air pressure is.  The change here is the change from normal atmospheric pressure to the pressure disturbance made by the sound.  This change in pressure can be measured by handheld devices or computers with microphones.

Sound pressure is usually measured in pascals, which is an SI (metric) unit.  A pascal (symbol Pa) is equal to a force of one newton per square meter.  A pascal is “small” compared to some of the pressure units one may be familiar with, such as pounds per square inch.  For instance, a tire pressure of 35 pounds per square inch is equal to about 241,000 pascals, or about 241 kilopascals. The smallest sound pressure a human ear can hear is 20 micropascals (0.000020 Pa).  Writing this is scientific notation is convenient: 2.0×10-5 Pa.  Remember this number – we will use this later as a reference.

The following table shows typical sound pressures of some recognizable things:

SourceSound Pressure (Pa)
Leaf rustling0.0000632
Normal conversation0.01
TV set at home0.02
Passenger car as heard from roadside0.1
Jack hammer2.0
Jet engine as heard from 100 yards100
Extremely loud rock band200
Jet engine as heard from 1 yard630

As you can see, these numbers range from very small ones, with four zeroes after the decimal point, to numbers in the hundreds. To make working with this range of numbers more manageable, a special logarithmic scale has been devised.

The Decibel

To refresh your memory, a logarithm function is the inverse of the exponent function. Examples of exponents are:

10= 100
10= 1000
10-1 = 0.1

The inverse of this, the logarithm function (log10), is as follows:

log10(100) = 2
log10(1000) = 3
log10(0.1) = -1

The last equation can be spoken as, “the log base 10 of 0.1 equals -1.” The decibel unit (symbol dB) is a logarithmic unit expressing the ratio between two values. The decibel was named in honor of the famous scientist Alexander Graham Bell (1847-1922). When measuring sound, we use the following logarithmic formula to determine the sound pressure level (SPL) in decibels.

Screen Shot 2015-02-26 at 11.13.16 AM

Here p is the sound pressure we are measuring, and Pref is our reference, the pressure of the smallest sound we can hear, 2.0×10-5 Pa (you remembered this, right?). For example, if we measure a sound pressure of 0.4 Pa, we can determine the SPL in dB:

formula

So a pressure change of 0.4 Pa equates to about 86 dB. In case you do not want to do the math on all of our examples above, here is that same table, now including sound pressure level.

SourceSound Pressure (Pa)Sound Pressure Level (dB)
Leaf rustling0.000063210
Normal conversation0.0154
TV set at home0.0260
Passenger car as heard from roadside0.174
Jack hammer2.0100
Jet engine as heard from 100 yards100134
Extremely loud rock band200140
Jet engine as heard from 1 yard630150

As you can see, we now have a range of numbers that seems more convenient – from 10 to 150 – instead of a range from a very, very, small number to a number in the hundreds. If you played around with the formula and plugged different pressure values in, you might notice that for every increase of 20 dB, the sound pressure has increased 10 times. That is because the scale with decibels is not linear, but rather logarithmic. That is a trade-off for using convenient numbers instead of the range from very small to very large ones.

Albert Einstein

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