Intensity
Discussion
intensity vs. amplitude
The amplitude of a sound wave can be quantified in several ways, all of which are a measure of the maximum change in a quantity that occurs when the wave is propagating through some region of a medium.
 Amplitudes associated with changes in kinematic quantities of the particles that make up the medium
 The displacement amplitude is the maximum change in position.
 The velocity amplitude is the maximum change in velocity.
 The acceleration amplitude is the maximum change in acceleration.
 Amplitudes associated with changes in bulk properties of arbitrarily small regions of the medium
 The pressure amplitude is the maximum change in pressure (the maximum gauge pressure).
 The density amplitude is the maximum change in density.
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? The velocity and acceleration changes caused by a sound wave are equally hard to measure in the particles that make up the medium.
Density fluctuations are minuscule and short lived. The period of a sound wave is typically measured in milliseconds. There are some optical techniques that make it possible to image the intense compressions are rarefactions associated with shock waves in air, but these are not the kinds of sounds we deal with in our everyday lives.
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. All types of amplitudes are equally valid for describing sound waves mathematically, but pressure amplitudes are the one we humans have the closest connection to.
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 telephone that connect to a microphone) the resulting value is called intensity. When done by a neuronal circuit (like the circuits in your brain that connect to your ears) the resulting sensation is called 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 dealt with at the end of this section, after the term level and its unit the decibel have been defined.
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 than 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 square 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. If you don't care to see the sausage being made below, jump to the equation just before the vibrant table.
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 up into two parts.





Mechanical waves in a continuous medium 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. Use the one dimensional wave equation for this.
∆s(x,t) = ∆s sin  ⎡ ⎢ ⎣ 
2π  ⎛ ⎜ ⎝ 
ft −  x  ⎞⎤ ⎟⎥ ⎠⎦ 
λ 
where…
∆s(x,t) =  instantaneous displacement at any position (x) and time (t) 
∆s =  displacement amplitude 
ƒ =  frequency 
λ =  wavelength 
π =  everyone's favorite mathematical constant 
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. In one period a wave would move forward one wavelength (λ).
m = ρV = ρAλ
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 start our calculation at any time we wish as long as we finish one cycle later. For convenience sake let's choose time to be zero — the beginning of a sinusoidal wave.





Clean up the constants.
½(ρA)(4π^{2}f^{2}∆s^{2}) = 2π^{2}ρAf^{2}∆s^{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 = ½λ 
λ  
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.
I = 2π^{2}ρf^{2}v∆s^{2}
We now have an equation that relates intensity (I) to displacement amplitude (∆s).
Does this formula make sense? Let's 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 ∝ f^{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 ∝ ∆s^{2}  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. 
Particle motion can be described in terms of displacement, velocity, or acceleration. Intensity can be related to these quantities as well. We've just completed the hard work of relating intensity (I) to displacement amplitude (∆s). For a sense of completeness (and for the sake of why not), let's also derive the equations for intensity in terms of velocity amplitude (∆v) and acceleration amplitude (∆a).
intensity and velocity
How does intensity relate to maximum velocity (the velocity amplitude)? Let's find out. Start with the one dimensional wave equation.
∆s(x,t) = ∆s sin  ⎡ ⎢ ⎣ 
2π  ⎛ ⎜ ⎝ 
ft −  x  ⎞⎤ ⎟⎥ ⎠⎦ 
λ 
Recall that velocity is the time derivative of displacement.



The stuff in front of the cosine function is the velocity amplitude.
∆v = 2πf∆s
Solve this for the displacement amplitude.
∆s =  ∆v 
2πf 
Just a little while ago, we derived an equation for intensity in terms of displacement amplitude.
I = 2π^{2}ρf^{2}v∆s^{2}
Combine these two equations…
I = 2π^{2}ρf^{2}v  ⎛ ⎜ ⎝ 
∆v  ⎞^{2} ⎟ ⎠ 
2πf 
and simplify.
I = ρv  ∆v^{2} 
2 
We now have an equation that relates intensity (I) to velocity amplitude (∆v).
intensity and acceleration
How does intensity relate to maximum acceleration (the acceleration amplitude)? Once again, let's find out. Once again, start with the one dimensional wave equation.
∆s(x,t) = ∆s sin  ⎡ ⎢ ⎣ 
2π  ⎛ ⎜ ⎝ 
ft −  x  ⎞⎤ ⎟⎥ ⎠⎦ 
λ 
Recall that velocity is the time derivative of displacement…



and that acceleration is the time derivative of velocity.



The acceleration amplitude is the stuff in front of the sine function (and ignoring the minus sign).
∆a = 4π^{2}f^{2}∆s
Rearrange this to make displacement amplitude the subject.
∆s =  ∆a 
4π^{2}f^{2} 
Time to bring back our equation for intensity in terms of displacement amplitude.
I = 2π^{2}ρf^{2}v∆s^{2}
Combine the previous two equations…
I = 2π^{2}ρf^{2}v  ⎛ ⎜ ⎝ 
∆a  ⎞^{2} ⎟ ⎠ 
4π^{2}f^{2} 
and simplify.
I = ρv  ∆a^{2} 
8π^{2}f^{2} 
We now have an equation that relates intensity (I) to acceleration amplitude (∆a).
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 intensitypressure equation from an effectively useless intensitydisplacement equation.
Start with the equation that relates intensity to displacement amplitude.
I = 2π^{2}ρf^{2}v∆s^{2}
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}f^{2}v^{2}∆s^{2} 
2ρv 
Write the numerator as a quantity squared.
I =  (2πρfv∆s)^{2} 
2ρv 
Look at the pile of symbols in the parenthesis.
2πρfv∆s
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 parenthetical quantity in the earlier equation is pressure — maximum gauge pressure to be more precise. We now have an equation that relates intensity to pressure amplitude.
I =  ∆P^{2} 
2ρv 
where…
I =  intensity [W/m^{2}] 
∆P =  pressure amplitude [Pa] 
ρ =  density [kg/m^{3}] 
v =  wave speed [m/s] 
Here's a slow and clean derivation of a the intensitypressure equation. Start from the version of Hooke's law that uses the bulk modulus (K).
F  = K  ∆V 
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 the same as a fractional change in length.
θ =  ∆V  =  ∂∆s(x,t) 
V_{0}  ∂x 
We have to use calculus here to get that fractional change, since the infinitesimal bits and pieces of the medium are squeezing and stretching at different rates at different points in space. Length changes are described by a onedimensional wave equation.
∆s(x,t) = ∆s sin  ⎡ ⎢ ⎣ 
2π  ⎛ ⎜ ⎝ 
ft −  x  ⎞⎤ ⎟⎥ ⎠⎦ 
λ 
Its spatial derivative is the same as the fractional change in volume.
θ =  ∂∆s(x,t)  = −  2π  ∆s cos  ⎡ ⎢ ⎣ 
2π  ⎛ ⎜ ⎝ 
ft −  x  ⎞⎤ ⎟⎥ ⎠⎦ 
∂x  λ  λ 
It's interesting to note that the volume changes are out of phase from the displacements, since taking the derivative changed sine to negative cosine. Volume changes are 90° behind displacement, since negative cosine is 90° behind sine. The most extreme volume changes occur at locations where the particles are back in their equilibrium 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. Doing that leaves us with this expression for the maximum strain (∆θ).
∆θ =  2π  ∆s 
λ 
Plugging this back into the bulk modulus equation gives us the maximum gauge pressure.
∆P = K  2π  ∆s 
λ 
And now for the dirty work. Recall these two equations for the speed of sound.
v = fλ  ⇒ 


⇒  K = v^{2}ρ 
Substitute into the previous equation…
∆P = v^{2}ρ  2πf  ∆s 
v 
and simplify.
∆P = 2πρfv∆s
Familiar? It's in the numerator of an expression that appeared earlier.
I =  (2πρfv∆s)^{2} 
2ρv 
Replace the pile of symbols in the parenthesis and behold. We get this thing again — the intensitypressure amplitude relationship.
I =  ∆P^{2} 
2ρv 
where…
I =  intensity [W/m^{2}] 
∆P =  pressure amplitude [Pa] 
ρ =  density [kg/m^{3}] 
v =  wave speed [m/s] 
intensity and density
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 pressure amplitudedisplacement amplitude relation…
∆P = 2πρfv∆s
substitute…
∆ρ =  2πρfv∆s 
v^{2} 
and simplify to get the densitydisplacement amplitude relation.
∆ρ =  2πρf∆s 
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} 
2ρv 
make a similar substitution…
I =  (∆ρv^{2})^{2} 
2ρv 
and simplify to get the equation that relates intensity to density amplitude.
I =  ∆ρ^{2}v^{3} 
2ρ 
Not very interesting, but now our list is complete.
amplitude  intensity  connection  

displacement 


velocity 

∆v = 2πf∆s  
acceleration 

∆a = 2πf∆v  
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 bel [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} = 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^{2} 
2ρv 
sound pressure level, SPL








text
L_{P} = 20 log  ⎛ ⎜ ⎝  ∆P  ⎞ ⎟ ⎠ 
∆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 a displacement 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 an approximate 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 5,000 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 intensity level (L) in neper [Np]. As with the bel it is customary to divide the neper into tenths or decineper [dNp]. The neper is also a dimensionless unit.



The neper and decineper are so rare in comparison to the bel and decibel that they are 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 
−24  sounds quieter than this are not possible due to the random motion of air molecules at room temperature (∆P = 1.27 μPa) 
−20.6  current world's quietest room (Microsoft Building 87, Redmond, Washinton) 
−9.4  former world's quietest room (Orfield Laboratories, Minneapolis, Minnesota) 
0  threshold of hearing, reference value for sound pressure (∆P_{0} = 20 μPa) 
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 
> 85  OSHA 1910.95(i)(1): Employers shall make hearing protectors available to all employees exposed to an 8hour timeweighted average of 85 decibels or greater at no cost to the employees. 
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 music player on high, power saw, rock concert, shouting in ear, symphony concert, video arcade, 
113  loudest clap (Alastair Galpin, New Zeeland, 2008) 
120–130  threshold of pain (∆P = 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 
125  loudest bird (white bellbird, Procnias albus) 
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, 1908 Tunguska meteor 
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)
 fish
 Unlike our ears and hydrophones, fish ears don't detect sound pressure, which is the compression of molecules. Instead, they perceive particle motion, the tiny backandforth movements of particles in response to sound waves.
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} = ⅔ 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.