Generally the loudness of a sound is related to the amplitude of the sound wave; a wave with bigger variations in pressure generally sounds louder. For any type of wave the energy carried by the wave is proportional to amplitude squared. This means doubling the amplitude increases the power by a factor of four (two squared). But the amount of energy reaching your ear also depends on the frequency since a wave with more oscillations per second (higher frequency) will mean the same amplitude hits your eardrum more often. Sound intensity is defined to be the energy per second (power in Watts) reaching a given area (measured in square meters). Normal conversation has an intensity of about 10^{-6} W/m^{2}.

A sound from a small sound source spreads out in all directions in an expanding spherical shape. Because the energy is spread over a larger and larger area as time goes on, the intensity decreases as you move further from the source. The area of a sphere is given by *4 π r ^{2}* so the decrease in intensity is proportional to

*r*where

^{2}*r*is the distance from the source. This is known as an inverse square law and many other laws in physics follow this same law. For example the gravitational field of the earth (or any other object) decreases as you move away from it proportional to the inverse square of the distance. So does the electric field around an electron or proton. For practical purposes, what this means is doubling the distance decreases the strength by 1/2

^{2}= 1/4. If you move three times as far away the strength is 1/3

^{2}or 1/9

^{th}as much. Shortening the distance by half means the intensity will be four times as much.

The human ear is an amazing instrument that can detect intensities as low as 10^{-12} W/m^{2} and can hear intensities as high as 10^{3} W/m^{2} (although this is loud enough to cause damage to the ear). To make this huge range easier to write down, a second scale of loudness was created called the sound intensity level, measured in decibels. The relationship between sound intensity, *I* measured in Watts per meter squared and sound intensity level (*SIL*) measured in decibels (dB), is given by *SIL = 10 log (I/I _{o})*. Here

*log*is the logarithm and

*I*10

_{o}=^{-12}W/m

^{2}is a reference sound intensity at about the threshold of human hearing.

Here are a few examples and rules of thumb for converting intensity (W/m^{2}) into intensity levels (in dB):

- A 10 fold increase in intensity equals an addition of 10dB. So going from a car horn to a jackhammer
*multiplies*the intensity by 10 (1 W/m^{2}to 10 W/m^{2}) but*adds*10 dB to the intensity level (120 dB to 130 dB). - A two fold increase in intensity (twice as loud in W/m
^{2}) equals and addition of 3 dB to the SIL. Suppose one trombone produces a sound level of 40dB. How loud are four trombones? Doubling the number of trombones to two adds 3dB, doubling again to four adds 3dB more so the new sound level is 46dB. - Suppose the sound intensity is 100 W/m
^{2}. What is the sound level? I = 10 log (100/10^{-12}) = 10 log (10^{14}) = 10*14 = 140 dB. - Suppose the sound level is 110 dB. What is the sound intensity? 110 dB = 10 log (I/10
^{-12}). Divide both sides by 10 to get 11 = log (I/10^{-12}). Now take inverse log 11 (same as 10^{11}) to get 10^{11}= I/^{-12}. Multiply both sides by^{-12}to get 0.1 W/m^{2}= I. - A Sound Conversion web site that converts between sound level, sound pressure and sound intensity.

Both sound intensity (W/m^{2}) and sound intensity level (SIL) are numbers that can be measured precisely in the laboratory (objective measurements). The human ear, however, is an imperfect measuring instrument. We hear better at a mid-range of frequencies than we do at very low or very high frequencies. The phon scale is a *subjective* measurement of loudness. This scale is arrived at by asking real humans to compare the loudness of different notes and an average is taken for many people (subjective). The units of the phon are the same as SIL units; the Decibels (dB).

The diagram below (modified from an MIT OpenCourseWare graph) relates sound intensity level (SIL, measured in dB with laboratory instruments), pressure (measured in W/m^{2} with laboratory instruments) and phons (human perception). A SIL of 110 dB is considered painful while a SIL of 0 is at the threshold of hearing. If our ears were the same as laboratory instruments the lines would go straight across. The phon scale and the SIL scale do give approximately the same number in dB but only for frequencies around 1000 Hz. In other words our subjective perception of loudness and the laboratory measurement agree but only for sounds with a frequency of 1000 Hz.

Notice there is a dip in all the curves between 1000 Hz and 5000 Hz indicating we are more sensitive to these frequencies and this is true for all loudness readings. For example suppose we perceive a sound at 4000 Hz to be 45 dB (phons) (labeled by a blue X in the diagram). The chart shows that at this loudness and frequency the dB reading in the laboratory is actually around 36 dB (dotted line to the SIL axis). So we perceive a sound of 36 dB (measured in the lab) as being much louder (45 dB) if it occurs at 4000 Hz. This is not surprising once you realize these are important frequencies for human speech; our hearing mechanism is built to hear human voices better than sound with much higher or much lower frequencies. This greater sensitivity around 3500 Hz is due to the tube resonance of the auditory canal (see chapter 12 for tube resonance and chapter 10 for a picture of the auditory canal).

It is also the case that intensity has an effect on perceived frequency; the same laboratory frequency will appear to be a slightly different frequency if the intensity is different. High frequencies are perceived to be a slightly higher pitch than normal if they are very loud. Low frequencies are perceived to be slightly lower than expected if they are very loud. Medium loudness doesn’t change the perceived pitch very much.

The above curves are parallel to the frequency response curves of microphones and speakers. No microphone has the same sensitivity to all frequencies and no speaker reproduces all frequencies equally well, as we will see in Chapter 18 on electronics. Likewise our hearing does not have the same sensitivity at all frequencies.