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 (the maximum particle displacement)
  2. by measuring the maximum change in density of the medium
  3. 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 time-averaged power (P) it transfers per area (A) 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 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
m2 m2

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
A
I =  E⟩/T
A
I =  K + Us⟩/T
A

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

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

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 the velocity of the particles in the medium (not the velocity of the wave through the medium).

v(x,t) =   Δx(x,t)
t
v(x,t) = 2πƒΔxmax cos

ft −  x ⎞⎤
⎠⎦
λ

Then square it.

v2(x,t) = 2ƒ2Δx2max 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
K = 
dK(x,0)
0
λ
K = 
1  (ρAdxv2(x,0)
2
0
λ
K = 
1  (ρA)(4π2ƒ2Δx2max)cos2
− 2π  x
 dx
2 λ
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 wave speed.)

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

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

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

One last bit of algebra and we're done. We now have an equation that relates intensity to displacement amplitude.

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 displacement 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

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 = ∆xmax sin(2π(ƒt −  x  + ϕ))
λ

Finding the amplitude of this equation is trivial.

xmax = ∆xmax

We just derived the equation that relates intensity to displacement amplitude.

I = 2π2ρƒ2vΔx2max

Velocity is the time derivative of displacement.

v =   ∆xmax sin(2π(ƒt −  x  + ϕ))
t λ
v = 2πƒ ∆xmax cos(2π(ƒt −  x  + ϕ))
λ

Find the amplitude…

vmax = 2πƒ∆xmax

…solve for xmax

xmax =  vmax
2πƒ

…substitute…

I = 2π2ρƒ2v
vmax 2
2πƒ

…and simplify.

I = ρv  v2max
2

Now we have an equation that relates intensity to velocity amplitude.

Accleration is the time derivative of velocity.

a =   2πƒ∆xmax cos(2π(ƒt −  x  + ϕ))
t λ
a = 4π2ƒ2xmax sin(2π(ƒt −  x  + ϕ))
λ

Find the amplitude…

amax = 4π2ƒ2xmax

…solve for xmax

xmax =  amax
2ƒ2

…substitute…

I = 2π2ρƒ2v
amax 2
2ƒ2

…and simplify.

I = ρv  a2max
2ƒ2

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 = 2π2ρƒ2vx2max

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 v.

I =  2ρ2ƒ2v2x2max
v

Write the numerator as a squared quantity.

I =  (2πρƒvxmax)2
v

Look at the pile of numbers and letters in the parenthesis in the numerator.

2πρƒvxmax

Look at the units of each physical quantity.


kg   1   m   m
m3 s s 1

Do some more magic — not algebra this time, but dimensional analysis.


kg  =  kg m  =  N  = Pa
m s2 m2 s2 m2

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 =  P2max
v

Where…

I =  intensity [W/m2]
Pmax =  pressure amplitude [Pa]
ρ =  density [kg/m3]
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 V0

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
V0 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 one-dimensional wave equation. Its spatial derivative is the same as the fractional change in volume.

x
 = ∆xmax sin(2π(ƒt −  x  + ϕ))
λ
∂∆x
 = −   ∆xmax cos(2π(ƒt −  x  + ϕ))
λ λ
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.

 ∆xmax
λ

Plugging this back into the bulk modulus equation gives us…

Pmax = K   ∆xmax
λ

And now for the dirty work. Recall these two equations for the speed of sound.

v = ƒλ  ⇒ 
1  =  ƒ
λ v
     
v = √ K
ρ
 ⇒  K = v2ρ

Substitute into the previous equation…

Pmax = v2ρ  2πƒ  ∆xmax
v

…and simplify to get the pressure-displacement amplitude relation.

Pmax = 2πρƒvxmax

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 Newton-Laplace 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
v2

Take the pressure-displacement amplitude relation…

Pmax = 2πρƒvxmax

…subsitute…

∆ρmax = 2πρƒvxmax
v2

…and simplify to get the density-displacement amplitude relation.

∆ρmax = 2πρƒ∆xmax
v

Mildly amusing. Let's try something else.

Again, assuming the first equation is the right one, solve it for P.

P = ∆ρv2

Take the equation that relates intensity to pressure amplitude…

I =  P2max
v

…make a similar substitution…

I =  (∆ρmaxv2)2
v

…and simplify to get the equation that relates intensity to density amplitude.

I =  ∆ρ2maxv3

Not very interesting, but now our list is complete.

Amplitude-Intensity Relationships v = wave speed, v = particle velocity
amplitude intensity connection
displacement
  I = 2π2ρƒ2vΔx2max
 
 
velocity
  I = ρv  v2max
  2
vmax = 2πƒ∆xmax
acceleration
  I = ρv  a2max
  2ƒ2
amax = 2πƒ∆vmax
pressure
  I =  P2max
  v
 
density
  I =  ∆ρ2maxv3
 
∆ρ =  P
v2

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.

Cosmo Kramer, 1991

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

LI[dB] = 10 log
I
I0

When the signal is a sound wave, this quantity is called the sound intensity level, frequently abbreviated SIL.

pressure

I =  Pmax2
v

pressure level

log
I
I0
 = log
(∆P2max)/(2ρv)
(∆P02)/(2ρv)
log
I
I0
 = log
Pmax 2
P0
log
I
I0
 = 2 log
Pmax
P0

text

LP[dB] = 20 log
Pmax
P0

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 (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.

LI[dNp] = 10 ln
I
I0
LP[dNp] = 20 ln
P
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.

level (dB) source
−∞ absolute silence
−9 world's quietest room (Orfield Labs, Minneapolis)
00–10 threshold of hearing (Pmax = 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 (Pmax = 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
Intensity level of selected sounds Sources: League for the Hard of Hearing and 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.