Discussion
welcome
The amplitude of a sound wave can be quantified in at least three ways:
 by measuring the maximum change in position of the particles that make up the medium (the maximum particle displacement)
 by measuring the maximum change in density of the medium
 by measuring the maximum change in pressure (the maximum gauge pressure)
Measuring displacement might as well be impossible. For typical sound waves, the maximum displacement of the molecules in the air is only a hundred or a thousand times larger than the molecules themselves — and what technologies are there for tracking individual molecules anyway?
Density fluctuations are equally minuscule and very short lived. (The period of sound waves is typically measured in milliseconds.) There are some optical techniques that make it possible to see the intense compressions are rarefactions associated with shock waves in air, but this will be dealt with in another section of this book.
Pressure fluctuations caused by sound waves are much easier to measure. Animals (including humans) have been doing it for several hundred million years with devices called ears. Humans have also been doing it electromechanically for about a hundred years with devices called microphones.
In any case, the results of such measurements are rarely ever reported. Instead, amplitude measurements are almost always used as the raw data in some computation. When done by an electronic circuit (like the circuits in a level meter) the resulting value is called the intensity. When done by a neuronal circuit (like the circuits in your brain) the resulting sensation is called the loudness.
The intensity of a sound wave is a combination of its rate and density of energy transfer. It is an objective quantity associated with a wave. Loudness is a perceptual response to the physical property of intensity. It is a subjective quality associated with a wave and is a bit more complex. As a general rule the larger the amplitude, the greater the intensity, the louder the sound. Sound waves with large amplitudes are said to be "loud". Sound waves with small amplitudes are said to be "quiet" or "soft". (The word "low" is sometimes also used to mean quiet, but this should be avoided. Use "low" to describe sounds that are low in frequency.) Loudness will be discussed at the end of this section.
By definition, the intensity (I) of any wave is the timeaveraged power (P) it transfers per area (A) through some region of space. The traditional way to indicate the timeaveraged value of a varying quantity is to enclose it in angle brackets ⟨⟩ . These look similar to the greater and less than symbols but they are taller and less pointy. That gives us an equation that looks like this…
I =  ⟨P⟩ 
A 
The SI unit of power is the watt, the SI unit of area is the suqare meter, so the SI unit of intensity is the watt per square meter — a unit that has no special name.
⎡ ⎣ 
W  =  W  ⎤ ⎦ 
m^{2}  m^{2} 
intensity and displacement
For simple mechanical waves like sound, intensity is related to the density of the medium and the speed, frequency, and amplitude of the wave. This can be shown with a long, horrible, calculation. Jump to the next highlighted equation if you don't care to see the sausage being made below.
Start with the definition of intensity. Replace power with energy (both kinetic and elastic) over time (one period, for convenience sake).





Since kinetic and elastic energies are always positive we can split the timeaveraged portion into two parts.





Mechanical waves in a continuous media can be thought of as an infinite collection of infinitesimal coupled harmonic oscillators. Little masses connected to other little masses with little springs as far as the eye can see. On average, half the energy in a simple harmonic oscillator is kinetic and half is elastic. The timeaveraged total energy in then either twice the average kinetic energy or twice the average potential energy.
⟨P⟩ =  2⟨K⟩  =  2⟨U_{s}⟩  
T  T 
Let's work on the kinetic energy and see where it takes us. It has two important parts — mass and velocity.
K = ½mv^{2}
The particles in a longitudinal wave are displaced from their equilibrium positions by a function that oscillates in time and space.
Δx(x,t) = Δx_{max} sin  ⎡ ⎣ 
2π  ⎛ ⎝ 
ft −  x  ⎞⎤ ⎠⎦ 
λ 
Take the time derivative to get the velocity of the particles in the medium (not the velocity of the wave through the medium).



Then square it.
v^{2}(x,t) = 
⎡ ⎣ 
2π  ⎛ ⎝ 
ft −  x  ⎞⎤ ⎠⎦ 
λ 
On to the mass. Density times volume is mass. The volume of material we're concerned with is a box whose area is the surface through which the wave is traveling and whose length is the distance the wave travels.
m = ρV = ρAx
In one period a wave would move forward one wavelength. In the volume spanned by a single wavelength, all the bits of matter are moving with different speeds. Calculus is needed to combine a multitude of varying values into one integrated value. We're dealing with a periodic system here, one that repeats itself over and over again. We can choose to do our calculation at any time we wish as long as we finish at the end of one cycle. For convenience sake let's choose time to be zero — the beginning of a sinusoidal wave.





Clean up the constants.
1  (ρA)(4π^{2}ƒ^{2}Δx^{2}_{max}) = 2π^{2}ρAƒ^{2}Δx^{2}_{max} 
2 
Then work on the integral. It may look hard, but it isn't. Just visualize the cosine squared curve traced out over one cycle. See how it divides the rectangle bounding it into equal halves?
The height of this rectangle is one (as in the number 1 with no units) and its width is one wavelength. That gives an area of one wavelength and a halfarea of half a wavelength.
λ  
⌠ ⌡ 
cos^{2}  ⎡ ⎣ 
− 2π  x  ⎤ ⎦ 
dx =  1  λ 
λ  2  
0 
Put the constants together with the integral and divide by one period to get the timeaveraged kinetic energy. (Remember that wavelength divided by period is wave speed.)



That concludes the hard part. Double the equation above and divide by area…



One last bit of algebra and we're done. We now have an equation that relates intensity to displacement amplitude.
I = 2π^{2}ρƒ^{2}vΔx^{2}_{max}
Does this formula make sense? Check to see how each of the factors affect intensity.
factor  comments 

I ∝ ρ  The denser the medium, the more intense the wave. That makes sense. A dense medium packs more mass into any volume than a rarefied medium and kinetic energy goes with mass. 
I ∝ ƒ^{2}  The more frequently a wave vibrates the medium, the more intense the wave is. I can see that one with my mind's eye. A lackluster wave that just doesn't get the medium moving isn't going to carry as much energy as one that shakes the medium like crazy. 
I ∝ v  The faster the wave travels, the more quickly it transmits energy. This is where you have to remember that intensity doesn't so much measure the amount of energy transferred as it measures the rate at which this energy is transferred. 
I ∝ Δx^{2}_{max}  The greater the displacement amplitude, the more intense the wave. Just think of ocean waves for a moment. A hurricanedriven, wallofwater packs a lot more punch than ripples in the bathtub. The metaphor isn't visually correct, since sound waves are longitudinal and ocean waves are complex, but it is intuitively correct. 
VERIFY EVERYTHING UP TO THE NEXT HEADING
Some people are interested in the way intensity relates to maximum velocity (the velocity amplitude) and maximum acceleration (the acceleration amplitude) as well as the maximum displacement (the displacement apmlitude). For sense of completeness, let's also derive these relationships.
Start with diplacement.
∆x = ∆x_{max} sin(2π(ƒt −  x  + ϕ)) 
λ 
Finding the amplitude of this equation is trivial.
∆x_{max} = ∆x_{max}
We just derived the equation that relates intensity to displacement amplitude.
I = 2π^{2}ρƒ^{2}vΔx^{2}_{max}
Velocity is the time derivative of displacement.
∆v =  ∂  ∆x_{max} sin(2π(ƒt −  x  + ϕ)) 
∂t  λ 
∆v = 2πƒ ∆x_{max} cos(2π(ƒt −  x  + ϕ))  
λ 
Find the amplitude…
∆v_{max} = 2πƒ∆x_{max}
…solve for ∆x_{max} …
∆x_{max} =  ∆v_{max} 
2πƒ 
…substitute…
I = 2π^{2}ρƒ^{2}v  ⎛ ⎝ 
∆v_{max}  ⎞^{2} ⎠ 
2πƒ 
…and simplify.
I = ρv  ∆v^{2}_{max} 
2 
Now we have an equation that relates intensity to velocity amplitude.
Accleration is the time derivative of velocity.
∆a =  ∂  2πƒ∆x_{max} cos(2π(ƒt −  x  + ϕ)) 
∂t  λ 
∆a = 4π^{2}ƒ^{2}∆x_{max} sin(2π(ƒt −  x  + ϕ)) 
λ 
Find the amplitude…
∆a_{max} = 4π^{2}ƒ^{2}∆x_{max}
…solve for ∆x_{max} …
∆x_{max} =  ∆a_{max} 
4π^{2}ƒ^{2} 
…substitute…
I = 2π^{2}ρƒ^{2}v  ⎛ ⎝ 
∆a_{max}  ⎞^{2} ⎠ 
4π^{2}ƒ^{2} 
…and simplify.
I = ρv  ∆a^{2}_{max} 
8π^{2}ƒ^{2} 
This completes the set with an equation that relates intensity to acceleration amplitude.
intensity and pressure
The amplitude of a sound wave can be measured much more easily with pressure (a bulk property of a material like air) than with displacement (the displacement of the submicroscopic molecules that make up air). Here's a quick and dirty derivation of a more useful intensity–pressure equation from an effectively useless intensity–displacement equation.
Start with the equation that relates intensity to displacement amplitude.
I = 2π^{2}ρƒ^{2}v∆x^{2}_{max}
Now let's play a little game with the symbols — a game called algebra. Note that many of the symbols in the equation above are squared. Make all of them squared by multiplying the numerator and denominator by 2ρv.
I =  4π^{2}ρ^{2}ƒ^{2}v^{2}∆x^{2}_{max} 
2ρv 
Write the numerator as a squared quantity.
I =  (2πρƒv∆x_{max})^{2} 
2ρv 
Look at the pile of numbers and letters in the parenthesis in the numerator.
2πρƒv∆x_{max}
Look at the units of each physical quantity.
⎡ ⎣ 
kg  1  m  m  ⎤ ⎦ 

m^{3}  s  s^{}  1 
Do some more magic — not algebra this time, but dimensional analysis.
⎡ ⎣ 
kg  =  kg m  =  N  = Pa  ⎤ ⎦ 
m s^{2}  m^{2} s^{2}  m^{2} 
The units of that mess are pascals, so the quantity in the numerator is pressure squared — maximum gauge pressure squared to be more precise. We now have an equation that relates intensity to pressure amplitude.
I =  ∆P^{2}_{max} 
2ρv 
Where…
I =  intensity [W/m^{2}] 
∆P_{max} =  pressure amplitude [Pa] 
ρ =  density [kg/m^{3}] 
v =  wave speed [m/s] 
The intensity of a sound wave depends not only on the pressure of the wave, but also on the density of the medium and speed of sound in the medium. Higher density and higher sound speed both give a lower intensity. Water is about 800 times more dense than air and has a speed of sound 4.5 times faster. Thus, sounds with the same pressure amplitude are about 3600 times more intense in air than in water. This is one of the reasons humans hear so poorly underwater. (The other reason is that our ears are really designed to work with air as the driving fluid, not water.)
Here's a slow and clean derivation of a the intensity–pressure equation. Start from the version of Hooke's law that uses the bulk modulus (K).
ƒ  = K  ΔV 
A  V_{0} 
The fraction on the left is the compressive stress, also known as the pressure (P). The fraction on the right is the compressive strain, also known as the fractional change in volume (θ). The latter of these two is the one we're interested in right now. Imagine a sound wave that only stretches and compresses the medium in one direction. If that's the case, then the fractional change in volume is effectively a fractional change in length.
θ =  ΔV  =  ∂∆x 
V_{0}  ∂x 
We have to use calculus here to get that fractional change, since the inifinitessimal bits and piece of the medium are squeezing and stretching at different rates at different points in space. Length changes are described by a onedimensional wave equation. Its spatial derivative is the same as the fractional change in volume.
∆x 


∂∆x 


∂x 
It's interesting to note that the volume changes are out of phase from the displacements. Volume changes are 90° behind displacement. The extreme volume changes occur at places where the particles are sitting at their original positions. Interesting, but not so useful right now. We care more about what these extreme values are than where they occur. For that, we replace the negative cosine expression with its extreme absolute value +1. This leaves us with this.
2π  ∆x_{max} 
λ 
Plugging this back into the bulk modulus equation gives us…
P_{max} = K  2π  ∆x_{max} 
λ 
And now for the dirty work. Recall these two equations for the speed of sound.
v = ƒλ  ⇒ 



⇒  K = v^{2}ρ 
Substitute into the previous equation…
P_{max} = v^{2}ρ  2πƒ  ∆x_{max} 
v 
…and simplify to get the pressuredisplacement amplitude relation.
∆P_{max} = 2πρƒv∆x_{max}
Familiar?
intensity and density
VERIFY EVERYTHING UP TO THE NEXT HEADING
The density changes in a medium associated with a sound wave are directly proportional to the pressure changes. The relationship is as follows…
v = √  ∆P 
∆ρ 
This looks similar to the NewtonLaplace equation for the speed of sound in an ideal gas but it's missing the heat capacity ratio γ (gamma). Why?
v = √  γP 
ρ 
Assuming the first equation is the right one, solve it for ∆ρ.
∆ρ =  ∆P 
v^{2} 
Take the pressuredisplacement amplitude relation…
∆P_{max} = 2πρƒv∆x_{max}
…subsitute…
∆ρ_{max} =  2πρƒv∆x_{max} 
v^{2} 
…and simplify to get the densitydisplacement amplitude relation.
∆ρ_{max} =  2πρƒ∆x_{max} 
v 
Mildly amusing. Let's try something else.
Again, assuming the first equation is the right one, solve it for ∆P.
∆P = ∆ρv^{2}
Take the equation that relates intensity to pressure amplitude…
I =  ∆P^{2}_{max} 
2ρv 
…make a similar substitution…
I =  (∆ρ_{max}v^{2})^{2} 
2ρv 
…and simplify to get the equation that relates intensity to density amplitude.
I =  ∆ρ^{2}_{max}v^{3} 
2ρ 
Not very interesting, but now our list is complete.
amplitude  intensity  connection  

displacement 


velocity 

∆v_{max} = 2πƒ∆x_{max}  
acceleration 

∆a_{max} = 2πƒ∆v_{max}  
pressure 


density 


levels
WRITE THIS PART
What is a level?
Types of levels.
I'm getting rid of all my furniture. All of it. And I'm going to build these different levels, with steps, and it'll all be carpeted with a lot of pillows. You know, like ancient Egypt.
Given a periodic signal of any sort, its intensity level (L_{I}) in bels [B] is defined as the base ten logarithm of the ratio of its intensity to the intensity of a reference signal. Since this unit is a bit large for most purposes, it is customary to divide the bel into tenths or decibels [dB]. The bel is a dimensionless unit.
L_{I}[dB] = 10 log  ⎛ ⎝ 
I  ⎞ ⎠ 
I_{0} 
When the signal is a sound wave, this quantity is called the sound intensity level, frequently abbreviated SIL.
pressure
I =  ∆P_{max}^{2} 
2ρv 
pressure level








text
L_{P}[dB] = 20 log  ⎛ ⎝ 
∆P_{max}  ⎞ ⎠ 
∆P_{0} 
Notes
 By convention, sound has a level of 0 dB at a pressure intensity of 20 μPa and frequency of 1,000 Hz. This is the generally agreed upon threshold of hearing for humans. Sounds with intensities below this value are inaudible to (quite possibly) every human.
 For sound in water and other liquids, a reference pressure of 1 μPa is used.
 The range of audible sound intensities is so great, that it takes six orders of magnitude to get us from the threshold of hearing (20 μPa ~ 0.5 pW/m^{2}) to the threshold of pain (20 Pa ~ 0.5 W/m^{2}).
 The bel was invented by engineers of the Bell telephone network in 1923 and named in honor of the inventor of the telephone, Alexander Graham Bell.
 A level of 0 dB is not the same as an intensity of 0 W/m^{2}, or a pressure amplitude of 0 Pa, or an amplitude of 0 m.
 Signals below the threshold or reference value are negative. Silence has a level of negative infinity.
 Since the base ten log of 2 is approximately 0.3, every additional 3 dB of level corresponds to a doubling of amplitude.
 A 10 decibel increase is perceived by people as sounding roughly twice as loud.
 Other examples of logarithmic scales include: earthquake magnitudes (often called by its obsolete name, the Richter scale), pH, stellar magnitudes, electromagnetic spectrum charts,… any more?
 Transform the decibel equation for level from a ratio to a difference.
 The 1883 eruption at Krakatau, Indonesia (often misspelled Krakatoa) had an intensity of 180 dB and was audible 5000 km away in Mauritius. The Krakatoa explosion registered 172 decibels at 100 miles from the source.
It would be equally reasonable to use natural logarithms in place of base ten, but this is far, far less common. Given a periodic signal of any sort, the ratio of the natural logarithm of its intensity to a reference signal is a measure of its level (L) in nepers [Np]. As with the bel it is customary to divide neper into tenths or decinepers [dNp]. The neper is also a dimensionless unit.



The neper and decineper are so rare in comparison to the bel and decibel that is essentially the answer to a trivia question.
Notes and quotes.
 Quote from Russ Rowlett of UNC: "The [neper] recognizes the British mathematician John Napier, the inventor of the logarithm. Napier often spelled his name Jhone Neper, and he used the Latin form Ioanne Napero in his writings." AHD "Scottish mathematician who invented logarithms and introduced the use of the decimal point in writing numbers."
 The value, in nepers, for the level difference of two values (F_{1} and F_{2}) of a field quantity is obtained by taking the natural logarithm of the ratio of the two values, ΔL_{N} = ln F_{1}/F_{2}. For socalled power quantities (see below), a factor 0.5 is included in the definition of the level difference, ΔL_{N} = 0.5 ln P_{1}/P_{2}. Two field quantity levels differ by 1 Np when the values of the quantity differ by a factor e (the base of natural logarithms). (The levels of two power quantities differ by 1 Np if the quantities differ by a factor e^{2}.) Since the ratio of values of any kind of quantity (or the logarithm of such ratios) are pure numbers, the neper is dimensionless and can be represented by "one." One cannot infer from this measure what kind of quantity is being considered so that the kind of quantity has to be specified clearly in all cases.
level (dB)  source 

−∞  absolute silence 
−9  world's quietest room (Orfield Labs, Minneapolis) 
00–10  threshold of hearing (∆P_{max} = 20 µPa), anechoic chamber 
10–20  normal breathing, rustling leaves 
20–30  whispering at 5 feet 
30–40  
40–50  coffee maker, library, quiet office, quiet residential area 
50–60  dishwasher, electric shaver, electric toothbrush, large office, rainfall, refrigerator 
60–70  air conditioner, automobile interior, alarm clock, background music, normal conversation, television, vacuum cleaner, washing machine 
70–80  coffee grinder, flush toilet, freeway traffic, hair dryer 
80–90  blender, doorbell, bus interior, food processor, garbage disposal, heavy traffic, hand saw, lawn mower, machine tools, noisy restaurant, toaster, ringing telephone, whistling kettle. Employers in the United States must provide hearing protectors to all workers exposed to continuous noise levels of 85 dB or above. 
090–100  electric drill, shouted conversation, tractor, truck 
100–110  baby crying, boom box, factory machinery, motorcycle, school dance, snow blower, snowmobile, squeaky toy held close to the ear, subway train, woodworking class 
110–120  ambulance siren, car horn, chain saw, disco, football game, jet plane at ramp, leaf blower, personal cassette player on high, power saw, rock concert, shouting in ear, symphony concert, video arcade, loudest clap (113 dB) 
120–130  threshold of pain (∆P_{max} = 20 Pa), auto stereo, band concert, chain saw, hammer on nail, heavy machinery, pneumatic drills, stock car races, thunder, power drill, percussion section at symphony 
130–140  air raid siren, jet airplane taking off, jackhammer 
140–150  
150–160  artillery fire at 500 feet, balloon pop, cap gun, firecracker, jet engine taking off 
160–170  fireworks, handgun, rifle 
170–180  shotgun 
180–190  rocket launch, 1883 Krakatau volanic eruption 
194  loudest sound possible in earth's atmosphere 
+∞  infinitely loud 
hearing
 loudness
 Loudness is a perceptual response to the physical property of intensity.
 A 10 dB increase in level is perceived by most listeners as a doubling in loudness
 A 1 dB change in level is just barely perceptible by most listeners
 Since loudness varies with frequency as well as intensity, a special unit has been designed for loudness — the phon. One phon is the loudness of a 1 dB, 1,000 Hz sound; 10 phon is the loudness of a 10 dB, 1,000 Hz sound; and so on.
 Cupping ones hand behind one's ear will result in an intensity increase of 6 to 8 dB.
 Asking someone to speak up usually results in an increase of about 10 dB on the part of the speaker.
 locating the source of sound
 Phase differences are one way we localize sounds. Only effective for wavelengths greater than 2 head diameters (eartoear distances). a.k.a. Interaural Time Difference (ITD)
 Sound waves diffract easily at wavelengths larger than the diameter of the human head (around 500 Hz wavelength equals 69 cm). At higher frequencies the head casts a "shadow". Sounds in one ear will be louder than the other. a.k.a. Interaural Level difference (ILD)
 The human ear can distinguish some…
 280 different intensity levels (seems unlikely)
seismic waves
Extended quote that needs to be paraphrased.
Magnitude scales are quantitative.With these scales, one measures the size of the earthquake as expressed by the seismic wave amplitude (amount of shaking at a point distant from the earthquake) rather than the intensity or degree of destructiveness. Most magnitude scales have a logarithmic basis, so that an increase in one whole number corresponds to an earthquake 10 times stronger than one indicated by the next lower number. This translates into an approximate 30fold increase in the amount of energy released. Thus magnitude 5 represents ground motion about 10 times that of magnitude 4, and about 30 times as much energy released. A magnitude 5 earthquake represents 100 times the ground motion and 900 times the energy released of a magnitude 3 earthquake.
The Richter scale was created by Charles Richter in 1935 at the California Institute of Technology. It was created to compare the size of earthquakes. One of Dr. Charles F. Richter's most valuable contributions was to recognize that the seismic waves radiated by all earthquakes can provide good estimates of their magnitudes. He collected the recordings of seismic waves from a large number of earthquakes, and developed a calibrated system of measuring them for magnitude. He calibrated his scale of magnitudes using measured maximum amplitudes of shear waves on seismometers particularly sensitive to shear waves with periods of about one second. The records had to be obtained from a specific kind of instrument, called a WoodAnderson seismograph. Although his work was originally calibrated only for these specific seismometers, and only for earthquakes in southern California, seismologists have developed scale factors to extend Richter's magnitude scale to many other types of measurements on all types of seismometers, all over the world. In fact, magnitude estimates have been made for thousands of moonquakes and for two quakes on Mars.
Most estimates of energy have historically relied on the empirical relationship developed by Beno Gutenberg and Charles Richter.
log_{10} E_{s} = 4.8 + 1.5 M_{s}
where energy, E_{s}, is expressed in joules. The drawback of this method is that M_{s} is computed from a bandwidth between approximately 18 to 22 s. It is now known that the energy radiated by an earthquake is concentrated over a different bandwidth and at higher frequencies. Note that this is not the total "intrinsic" energy of the earthquake, transferred from sources such as gravitational energy or to sinks such as heat energy. It is only the amount radiated from the earthquake as seismic waves, which ought to be a small fraction of the total energy transferred during the earthquake process.
With the worldwide deployment of modern digitally recording seismograph with broad bandwidth response, computerized methods are now able to make accurate and explicit estimates of energy on a routine basis for all major earthquakes. A magnitude based on energy radiated by an earthquake, M_{e}, can now be defined. These energy magnitudes are computed from the radiated energy using the Choy and Boatwright (1995) formula
M_{e} = (2/3) log_{10} E_{s} − 2.9
where E_{s} is the radiated seismic energy in joules. M_{e}, computed from high frequency seismic data, is a measure of the seismic potential for damage.