Decibels, Sound Intensity and Frequency Response

It was mentioned in previous articles that the sensitivity of the human ear is phenomenally keen. The threshold of hearing (what a young perfect ear could hear) is 1 x 10^-12Watts /meter^2. This way of expressing sound wave amplitude is referred to as Sound Intensity (I). It is not to be confused with Sound Intensity Level (L), measured in decibels (dB). The reason why loudness is routinely represented in decibels rather than Watts/meter^2 is primarily because the ears don’t hear linearly. That is, if the sound intensity doubles, it doesn’t sound twice as loud. It doesn’t really sound twice as loud until the Sound Intensity is about ten times greater. (This is a very rough approximation that depends on the frequency of the sound as well as the reference intensity.) If the sound intensity were used to measure loudness, the scale would have to span 14 orders of magnitude. That means that if we defined the faintest sound as “1”, we would have to use a scale that went up to 100,000,000,000,000 (about the loudest sounds you ever hear. The decibel scale is much more compact (0 dB – 140 dB for the same range) and it is more closely linked to our ears’ perception of loudness. You can think of the sound intensity as a physical measure of sound wave amplitude and sound intensity level as its psychological measure.
The equation that relates sound intensity to sound intensity level is:


L = 10 log(I2/I1)


L ≡ The number of decibels I2 is greater than I1;
I2 ≡ The higher sound intensity being compared;
I1 ≡ The lower sound intensity being compared;


Remember, I is measured in Watts /meter^2. It is like the raw power of the sound. The L in this equation is what the decibel difference is between these two. In normal use, I1 is the threshold of hearing, 1 x 10^-12Watts /meter^2. This means that the decibel difference is with respect to the faintest sound that can be heard. So when you hear that a busy intersection is 80 dB, or a whisper is 20 dB, or a class cheer is 105 dB, it always assumes that the comparison is to the threshold of hearing, 0 dB. (“80 dB” means 80 dB greater than threshold). Don’t assume that 0 dB is no sound or total silence. This is simply the faintest possible sound a human with perfect hearing can hear. The Table shown on Picture 1 provides decibel levels for common sounds.

Picture 1: Decibel levels for typical sounds



If you make I2 twice as large as I1, then ΔL = 3 dB. If you make I2 ten times as large as I1, then ΔL = 10 dB. These are good reference numbers to tuck away:


Double Sound Intensity ~ +3 dB

10 x Sound Intensity = +10 dB

Frequency Response over the Audible Range

We hear lower frequencies as low pitches and higher frequencies as high pitches. However, our sensitivity varies tremendously over the audible range. For example, a 50 Hz sound must be 43 dB before it is perceived to be as loud as a 4,000 Hz sound at 2 dB. (4,000 Hz is the approximate frequency of greatest sensitivity for humans with no hearing loss.) In this case, we require the 50 Hz sound to have 13,000 times the actual intensity of the 4,000 Hz sound in order to have the same perceived intensity! The Table shown on Picture 2 illustrates this phenomenon of sound intensity level versus frequency. The last column puts the relative intensity of 4,000 Hz arbitrarily at 1 for easy comparison with sensitivity at other frequencies.

Picture 2: Sound intensity and sound intensity level required to perceive sounds at different frequencies to be equally loud


Another factor that affects the intensity of the sound you hear is how close you are to the sound. Obviously a whisper, barely detected at one meter could not be heard across a football field. Here’s the way to think about it. The power of a particular sound goes out in all directions. At a meter away from the source of sound, that power has to cover an area equal to the area of a sphere (4πr^2) with a radius of one meter. That area is 4π m^2. At two meters away the same power now covers an area of 4π(2 m)2 = 16π m^2, or four times as much area. At three meters away the same power now covers an area of 4π(3 m)2 = 36π m^2, or nine times as much area. So compared to the intensity at one meter, the intensity at two meters will be only one-quarter as much and the intensity at three meters only one-ninth as much. The sound intensity follows an inverse square law, meaning that by whatever factor the distance from the source of sound changes, the intensity will change by the square of the reciprocal of that factor.

Another way to look at this is to first consider that the total power output of a source of sound is its sound intensity in Watts/meter^2 multiplied by the area of the sphere that the sound has reached. So, for example, the baby in the problem above creates a sound intensity of 1x10^-2 W/m^2 at 2.5 m away. This means that the total power put out by the baby is: Power = Intensity x sphere area -> P = 0.785 W.

Now, if we calculate the power output at 6 meters away, where the intensity is 1.74 x 10^-3 W/m^2, we will get the same result -> P = 0.785 W. The result is same, because the power output depends on the baby, not the position of the observer. This means we can always equate the power outputs that are measured at different locations:

P1 = P2 -> I1(4πr1^2) = I2(4πr2^2)

-> I1r1^2 = I2r2^2

Sound Characteristics

All sound waves are compressional waves caused by vibrations, but the music from a symphony varies considerably from both a baby’s cry and the whisper of a confidant. All sound waves can be characterized by their speed, by their pitch, by their loudness, and by their quality or timbre.

The Speed of Sound

The speed of sound is fastest in solids (almost 6000 m/s in steel), slower in liquids (almost 1500 m/s in water), and slowest in gases. We normally listen to sounds in air, so we’ll look at the speed of sound in air most carefully. In air, sound travels at:


v = 331 [m/s] + 0.6 [m/s]/[°C] * T [°C]


The part to the right of the “+” sign is the temperature factor. It shows that the speed of sound increases by 0.6 m/s for every temperature increase of 1°C. So, at 0° C, sound travels at 331 m/s (about 740 mph). But at room temperature (about 20°C) sound travels at:


v = 343 [m/s]


It was one of aviation’s greatest accomplishments when Chuck Yeager, on Oct. 14, 1947, flew his X-1 jet at Mach 1.06, exceeding the speed of sound by 6%. Regardless, this is a snail’s pace compared to the speed of light. Sound travels through air at about a million times slower than light, which is the reason why we hear sound echoes but don’t see light echoes. It’s also the reason we see the lightning before we hear the thunder. The lightning-thunder effect is often noticed in big stadiums. If you’re far away from a baseball player who’s up to bat, you can clearly see the ball hit before you hear the crack of the bat. You can consider that the light recording the event reaches your eyes virtually instantly. So if the sound takes half a second more time than the light, you’re half the distance sound travels in one second (165 meters) from the batter. Next time you’re in a thunderstorm use this method to estimate how far away lightning is striking.

The Pitch of Sound

The pitch of sound is the same as the frequency of a sound wave. With no hearing losses or defects, the ear can detect wave frequencies between 20 Hz and 20,000 Hz. (Sounds below 20 Hz are classified as subsonic; those over 20,000 Hz are ultrasonic). However, many older people, loud concert attendees, and soldiers with live combat experience lose their ability to hear higher frequencies. The good news is that the bulk of most conversation takes place well below 2,000 Hz. The average frequency range of the human voice is 120 Hz to approximately 1,100 Hz (although a baby’s shrill cry is generally 2,000 - 3,000 Hz – which is close to the frequency range of greatest sensitivity). Even telephone frequencies are limited to below 3,400 Hz. But the bad news is that the formation of many consonants is a complex combination of very high frequency pitches. So to the person with high frequency hearing loss, the words key, pee, and tea sound the same. You find people with these hearing losses either lip reading or understanding a conversation by the context as well as the actual recognition of words.

One important concept in music is the octave – a doubling in frequency. For example, 40 Hz is one octave higher than 20 Hz. The ear is sensitive over a frequency range of about 10 octaves: 20 Hz -> 40 Hz -> 80 Hz -> 160 Hz -> 320 Hz -> 640 Hz -> 1,280 Hz -> 2,560 Hz -> 5,120 Hz -> 10,240 Hz -> 20,480 Hz. And within that range it can discriminate between thousands of differences in sound frequency. Below about 1,000 Hz the Just Noticeable Difference (JND) in frequency is about 1 Hz (at the loudness at which most music is played), but this rises sharply beyond 1,000 Hz. At 2,000 the JND is about 2 Hz and at 4,000 Hz the JND is about 10 Hz. (A JND of 1 Hz at 500 Hz means that if you were asked to listen alternately to tones of 500 Hz and 501 Hz, the two could be distinguished as two different frequencies, rather than the same). It is interesting to compare the ear’s frequency perception to that of the eye. From red to violet, the frequency of light less than doubles, meaning that the eye is only sensitive over about one octave, and its ability to discriminate between different colors is only about 125. The ear is truly an amazing receptor, not only its frequency range, but also in its ability to accommodate sounds with vastly different loudness.

The Loudness of Sound

The loudness of sound is related to the amplitude of the sound wave. Most people have some recognition of the decibel (dB) scale. They might be able to tell you that 0 dB is the threshold of hearing and that the sound on the runway next to an accelerating jet is about 140 dB. However, most people don’t realize that the decibel scale is a logarithmic scale. This means that for every increase of 10 dB the sound intensity increases by a factor of ten. So going from 60 dB to 70 dB is a ten-fold increase, and 60 dB to 80 dB is a hundred-fold increase. This is amazing to me. It means that we can hear sound intensities over 14 orders of magnitude. This means that the 140 dB jet on the runway has a loudness of 10^14 times greater than threshold. 10^14 is 100,000,000,000,000 – that’s 100 trillion! It means our ears can measure loudness over a phenomenally large range. Imagine having a measuring cup that could accurately measure both a teaspoon and 100 trillion teaspoons (about 10 billion gallons). The ear is an amazing receptor! However, our perception is skewed a bit. A ten-fold increase in loudness doesn’t sound ten times louder. It may only sound twice as loud. That’s why when cheering competitions are done at school rallies, students are not very excited by the measure of difference in loudness between a freshmen class (95 dB) and a senior class (105 dB). The difference is only 10 dB. It sounds perhaps twice as loud, but it’s really 10 times louder.

The Quality of Sound

The quality of sound or timbre is the subtlest of all its descriptors. A trumpet and a violin could play exactly the same note, with the same pitch and loudness, and if your eyes were closed you could easily distinguish between the two. The difference in the sounds has to do with their quality or timbre. The existence of supplementary tones, combined with the basic tones, doesn’t change the basic pitch, but gives a special “flavor” to the sound being produced. Sound quality is the characteristic that gives the identity to the sound being produced.

Sound, Sound waves and Ear

We already had an basic approach on sound and sound waves in this article: Sound Waves. Now we will extend this approach into more details, in order to understand the basic sound theory and the basic functionality of our sound detector, the ear. Our ears are incredibly awesome receptors for sound waves. The threshold of hearing is somewhere around 1x10^-12 W/m^2. At this smallest of perceptible sound intensities, the eardrum vibrates less distance than the diameter of a hydrogen atom! Well, it's so small an amount of power that you can hardly conceive of it, but if you have pretty good hearing, you could detect a sound wave with that small amount of power. That's not all. You could also detect a sound wave a thousand times more powerful, a million times more powerful, and even a billion times more powerful. And, that's before it even starts to get painful!

Types of Sound Waves

Waves come in two basic types, depending on their type of vibration. Imagine laying a long slinky on the ground and shaking it back and forth. You would make what is called a transverse wave (see Picture 1 (a)). A transverse wave is one in which the medium vibrates at right angles to the direction the energy moves. If instead, you pushed forward and pulled backward on the slinky you would make a compressional wave (see Picture 1 (b)). Compressional waves are also known as longitudinal waves. A compressional wave is one in which the medium vibrates in the same direction as the movement of energy in the wave.

Picture 1: Transverse and Longitudinal (compressional) wave


If we take a snapshot of one transverse wave from the ceiling, it would look like Picture 2.

Picture 2: Wave elements


CREST: The topmost point of the wave medium or greatest positive distance from the rest position.

TROUGH: The bottom most point of the wave medium or greatest negative distance from the rest position.

WAVELENGTH (λ): The distance from crest to adjacent crest or from trough to adjacent trough or from any point on the wave medium to the adjacent corresponding point on the wave medium.

AMPLITUDE (A): The distance from the rest position to either the crest or the trough. The amplitude is related to the energy of the wave. As the energy grows, so does the amplitude. This makes sense if you think about making a more energetic slinky wave. You’d have to swing it with more intensity, generating larger amplitudes. The relationship is not linear though. The energy is actually proportional to the square of the amplitude. So a wave with amplitude twice as large actually has four times more energy and one with amplitude three times larger actually has nine times more energy.

FREQUENCY (f): The number of wavelengths to pass a point per second. In this case the frequency would be 3 per second. Wave frequency is often spoken of as “waves per second,” “pulses per second,” or “cycles per second.” However, the SI unit for frequency is the Hertz (Hz). 1 Hz = 1 per second, so in the case of this illustration, f = 3 Hz.

PERIOD (T): The time it takes for one full wavelength to pass a certain point. If you see three wavelengths pass your foot every second, then the time for each wavelength to pass is 1/3 of a second. Period and frequency are reciprocals of each other:

T = 1/f 
f = 1/T

SPEED (v): Average speed is always a ratio of distance to time, v = d /t . In the case of wave speed, an appropriate distance would be wavelength, λ. The corresponding time would then have to be the period, T. So the wave speed becomes:

v = λ/T 
v = λf

Sound Waves

If a tree falls in the forest and there’s no one there to hear it, does it make a sound? It’s a common question that usually evokes a philosophical response. I could argue yes or no convincingly. You will too later on. Most people have a very strong opinion one way or the other. The problem is that their opinion is usually not based on a clear understanding of what sound is.
One of the easiest ways to understand sound is to look at something that has a simple mechanism for making sound. Think about how a tuning fork makes sound. Striking one of the forks causes you to immediately hear a tone. The tuning fork begins to act somewhat like a playground swing. The playground swing, the tuning fork, and most physical systems will act to restore themselves if they are stressed from their natural state. The “natural state” for the swing, is to hang straight down. If you push it or pull it and then let go, it moves back towards the position of hanging straight down. However, since it’s moving when it reaches that point, it actually overshoots and, in effect, stresses itself. This causes another attempt to restore itself and the movement continues back and forth until friction and air resistance have removed all the original energy of the push or pull. The same is true for the tuning fork. It’s just that the movement (amplitude) is so much smaller that you can’t visibly see it. But if you touched the fork you could feel it. Indeed, every time the fork moves back and forth it smacks the air in its way. That smack creates a small compression of air molecules that travels from that point as a compressional wave. When it reaches your ear, it vibrates your eardrum with the same frequency as the frequency of the motion of the tuning fork. You mentally process this vibration as a specific tone.

Picture 3: The front of a speaker cone faces upward with several pieces of orange paper lying on top of it


A sound wave is nothing more than a compressional wave caused by vibrations. Next time you have a chance, gently feel the surface of a speaker cone (see Picture 3). The vibrations you feel with your fingers are the same vibrations disturbing the air. These vibrations eventually relay to your ears the message that is being broadcast. So, if a tree falls in the forest and there’s no one there to hear it, does it make a sound? Well … yes, it will certainly cause vibrations in the air and ground when it strikes the ground. And … no, if there’s no one there to mentally translate the vibrations into tones, then there can be no true sound. You decide. Maybe it is a philosophical question after all.