Intensity

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© 1998-2008 by Glenn Elert -- A Work in Progress
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welcome

The amplitude of a sound wave can be quantified in at least three ways:

  1. by measuring the maximum change in position of the particles that make up the medium,
  2. by measuring the maximum change in density of the medium, or
  3. by measuring the maximum change in pressure (the maximum gauge pressure).

None of these quantities are used much, however. In fact, the first two are unusually difficult to measure directly. 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. Any resulting density fluctuations are equally miniscule and very short lived. (The period of sound waves is typically measured in milliseconds.)

Pressure fluctuations caused by sound waves are much easier to measure (animals have been doing it for hundreds of millions of years with ears), but 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.

Briefly, 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 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 the word "low" only to describe sounds that are low in frequency or pitch.) Loudness will be discussed at the end of this section.

By definition, the intensity of any wave is the time-averaged rate at which it transmits power per unit area through some region of space. One way to indicate the time averaged value of a varying quantity is to enclose it in angle brackets. (These look similar to greater and less than symbols but are taller and less pointy.) That gives us an equation that looks something like this …

I =  〈P〉
A

The unit of intensity is the watt per square meter — a unit that has no special name.

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, calculus derivation.

Start with the definition of intensity. Then replace power with energy (in the case of mechanical waves, that means kinetic energy) over time (one period, for convenience sake).

I =  〈P〉  =  〈K〉 / T
A A

The difficult part of this derivation is determining the time-averaged value of the kinetic energy. We need to integrate the kinetic energy over one period. When we do this, we actually get the time-average value multiplied by one period. Dividing the period out cures that problem. Here is what I have been talking about represented in mathematical symbols.

    T        
〈K〉 =  ⎰
⎱		 ⌠
⌡		 1  mv2 dt ⎱
⎰		  ÷ T
2
    0        

Let's work out the details of the interior of the integral. The particles in a longitudinal wave are displaced from their equilibrium positions by a function that oscillates in time and space.

∆x(x,t) = ∆xmax sin  ⎡
⎣		  2π  ⎛
⎝		  ft −  x ⎞⎤
⎠⎦
λ

Take the time derivative to get velocity.

v(x,t) =  ∂  ∆x(x,t) = 2πf∆xmax cos  ⎡
⎣		  2π  ⎛
⎝		  ft −  x ⎞⎤
⎠⎦
∂t λ

Then square it.

v(x,t)2 = 4π2f2∆x2max cos2  ⎡
⎣		  2π  ⎛
⎝		  ft −  x ⎞⎤
⎠⎦
λ

That's the big part of the interior of the integral. The smaller part is the mass. For continuous systems, like the media through which a sound wave passes, mass doesn't make as much sense as density. Still, the kinetic energy formula demands a mass. The volume of material in motion is a box whose area is the surface through which the wave is travelling and whose length is one wavelength. Here's what those words look like as symbols …

m = ρV = ρλA

Back to the integral. To keep from getting messy, I recommend pulling all the constant terms out right away.

T         T                
⌠
⌡		 1  mv2 dt =  (ρλA) ( 4π2f2∆x2max)   ⌠
⌡		  cos2  ⎡
⎣		  2π  ⎛
⎝		  ft −  x ⎞⎤
⎠⎦		  dt
2 2 λ
0         0                

Clean up the constants.

(ρλA) (4π2f2∆x2max)  = 2π2λρAf2∆x2max
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 two equal halves?

The height of this rectangle is one (as in, the number 1) and its width is one period. That gives an area of one period and a half-area of half a period.

T                    
⌠
⌡		  cos2  ⎡
⎣		  2π  ⎛
⎝		  ft −  x ⎞⎤
⎠⎦		  dt =  1  T
λ 2
0                    

Put the constants together with the integral and divide by one period to get the time-averaged kinetic energy.

  〈K〉 =  ⎛
⎝		 2π2λρAf2∆x2max ⎞⎛
⎠⎝		 1  T ⎞
⎠		  ÷ T = π2λρAf2∆x2max
2

That concludes the hard part. Now go back to the begining of this derivation and substitute. (Recall that period and frequency are reciprocals.)

I =  〈P〉  =  〈K〉 / T  =  〈K〉 f  =  (π2λρAf2∆x2max) f
A A A A

And then finish it off. (This time, recall that frequency times wavelength is speed.)

I = π2ρf2v∆x2max

[The answer should be … I = 2π2ρf2v∆x2max]

Does this formula make sense? Check to see how each of the factors affect intensity.

Factors Affecting the Intensity of Sound Waves
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 ∝ f2   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 ∝ ∆x2max The greater the amplitude, the more intense the wave. Just think of ocean waves for a moment. A hurricane-driven, wall-of-water 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.

intensity and pressure

Don't forget to breathe, then discuss intensity and pressure amplitude. Amplitude is measured in meters [m] while pressure amplitude is measured in pascals [Pa] or more commonly millipascals [mPa] or most commonly micropascals [μPa]. Intensity is proportional to the square of pressure amplitude.

I =  〈Power〉  =  〈Fv〉  = 〈Pressure v〉
A A

For simple harmonic motion …

vmax = 2πf∆xmax

Dimensional analysis game …

intensity and density

 … 

power level

The range of audible sound intensities is so great, that they span six orders of magnitude from the threshold of hearing (20 μPa) to the threshold of pain (20 Pa).

Given a periodic signal of any sort, its level (β) 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.

β[dB] = 10 log  ⎛
⎝		 I ⎞
⎠
I0
log  ⎛
⎝		 I ⎞
⎠		  = log  ⎛
⎜
⎝		 (P2) / (2ρv) ⎞
⎟
⎠		  = log  ⎛
⎝		 P ⎞2
⎠		  = 2 log  ⎛
⎝		 P ⎞
⎠
I0 (P02) / (2ρv) P0 P0
β[dB] = 20 log  ⎛
⎝		 P ⎞
⎠
P0

By convention, sound has a level of 0 dB at a pressure intensity of 20 μPa and frequency of 1000 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.

Notes

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 (β) 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.

β[dNp] = 10 ln  ⎛
⎝		 I ⎞
⎠		  = 20 ln  ⎛
⎝		 P ⎞
⎠
I0 P0

The neper and decineper are so rare in comparison to the bel and decibel that is essentially the answer to a trivia question. Quote from Russ Rowlett of UNC: "The [neper] recognizes the British mathematician John Napier (1550-1617), 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."

Intensity of Selected Sounds
intensity (dB) source
000 - 010 threshold of hearing, anechoic chamber
010 - 020 normal breathing, rustling leaves
020 - 030 whispering at 5 feet
030 - 040  
040 - 050 coffee maker, library, quiet office, quiet residential area
050 - 060 dishwasher, electric shaver, electric toothbrush, large office, rainfall, refrigerator, sewing machine
060 - 070 air conditioner, automobile interior, alarm clock, background music, normal conversation, television, vacuum cleaner, washing machine
070 - 080 coffee grinder, flush toilet, freeway traffic, garbage disposal, hair dryer
080 - 090 blender, doorbell, bus interior, food processor, garbage disposal, heavy traffic, hand saw, lawn mower, machine tools, noisy restaurant, toaster, ringing telephone, whistling kettle
OSHA regulations mandate hearing protection for continuous occupational exposure to intensities over 85 dB.
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
120 - 130 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 threshold of pain, 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, volcanic eruption
190 - 200  
Sources: League for the Hard of Hearing, Physics of the Body

hearing

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 30-fold 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 Wood-Anderson 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.

log10 Es = 4.8 + 1.5 Ms

where energy, Es, is expressed in joules. The drawback of this method is that Ms 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, Me, can now be defined. These energy magnitudes are computed from the radiated energy using the Choy and Boatwright (1995) formula

Me = (2/3) log10 Es - 2.9

where Es is the radiated seismic energy in joules. Me, computed from high frequency seismic data, is a measure of the seismic potential for damage.

Summary

Problems

practice

  1. Write something.
    • Answer it.
  2. Write something else.
    • Answer it.
  3. Write something different.
    • Answer it.
  4. Write something completely different.
    • Answer it.

numerical

  1. Determine the maximum displacement of a gas molecule in the air from a 1000 Hz tone with an intensity of …
    1. 1 pW/m2 (threshold of sensation)
    2. 1 W/m2 (threshold of pain).

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