Intensity

Discussion

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 power it transfers per area through some region of space. The traditional 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, 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).

I =  P  =  E⟩/T  =  K + Us⟩/T
A A A

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

P⟩ =  E  =  K + Us  =  K  +  Us
T T T T

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 time-averaged total energy in then either twice the average kinetic energy or twice the average potential energy.

P⟩ =  2⟨K  =  2⟨Us
T T

Let's work on the kinetic energy and see where it takes us. It has two important parts — mass and speed.

K = ½mv2

The particles in a longitudinal wave are displaced from their equilibrium positions by a function that oscillates in time and space.

Δx(x,t) = Δxmax sin 

ft −  x ⎞⎤
⎠⎦
λ

Take the time derivative to get velocity.

v(x,t) =   Δx(x,t) = 2πƒΔxmax cos 

ft −  x ⎞⎤
⎠⎦
t λ

Then square it.

v2(x,t) = 4π2ƒ2Δx2max cos2 

 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.

λ λ λ
K = 
dK(x,0) = 
1  (ρAdxv2(x,0) = 
1  (ρA)(4π2ƒ2Δx2max) cos2 
− 2π  x
 dx
2 2 λ
0 0 0

Clean up the constants.

1  (ρA)(4π2ƒ2Δx2max) = 2π2ρAƒ2Δx2max
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 half-area of half a wavelength.

λ

cos2 
− 2π  x
dx =  1  λ
λ 2
0

Put the constants together with the integral and divide by one period to get the time-averaged kinetic energy. (Remember that wavelength divided by period is speed.)

K  =
(2π2ρAƒ2Δx2max)( 1 λ)
1  = π2ρAƒ2vΔx2max
T 2 T

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

I =  P  =  2⟨K⟩/T  =  2(π2ρAƒ2vΔx2max)
A A A

… and we're done.

I = 2π2ρƒ2vΔx2max

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 ∝ ƒ2 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 ∝ Δx2max 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.
Factors affecting the intensity of sound waves

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  = ⟨pressurev
A A

For simple harmonic motion …

vmax = 2πƒΔxmax

Dimensional analysis game …

intensity and density

 … 

power level

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) 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.

Notes and quotes.

intensity (dB) source
−∞ absolute silence
−9 world's quietest room (Orfield Labs, Minneapolis)
00–10 threshold of hearing, 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, sewing machine
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, garbage disposal, 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 Alistair Galpin)
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  
+∞ infinitely loud
Intensity of selected sounds Sources: League for the Hard of Hearing andPhysics 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.