PES 2130 Fall 2014, Spendier
Lecture 12/Page 1
Lecture today: Chapter 17 Waves-2
1) Sound Waves
2) Traveling Sound Waves
3) The Speed of Sound
4) Intensity and Sound Level
Announcements:
- HW 4 due, HW 5 given out
- Due to travel, no office hours this week Thursday
DEMO:
The Rubens' Flame Tube: Seeing Sound through Fire
http://www.youtube.com/watch?v=ynqzeIYA7Iw
Seeing Sound Through Waves:
A speaker pumps sound waves into a long pipe. The sound wave reflects back and forth
inside pipe creating a resultant wave (the superposition), which for particular frequencies
creates standing waves (resonance). Propane gas is pumped into the pipe, and leaks out a
series of holes drilled on the top side, creating a wall of flame. When a standing wave is
present inside the tube, the height of the flames reveals the unseen nodes and antinodes
inside the pipe.
1) Pressure varies along pipe when we put a sound wave in
2) Where pressure is largest, gas is pushed out the most
3) By setting the gas on fire, we can see these maximum pressure positions, as this is
where the flame is highest (antinodes of standing sound wave)
Pyro Board: 2D Rubens' Tube
https://www.youtube.com/watch?v=2awbKQ2DLRE
1) Sound Waves
Examples:
- Dolphins emit sound waves (~ 100,000 Hz) in water and use echo for guidance and
hunting
- Ultrasound imaging are sound waves that are scanned over the body and the "echoes"
from internal organs are used to create an image (5 MHz)
most general definition: longitudinal waves in a medium (net motion of molecules is in
the same direction (parallel) that the wave propagates
- no sound: molecules of medium are uniformly distributed
- sound: localized regions of increased/decreased pressure & density of molecules in
medium
similar to compression
waves in a slinky!
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Lecture 12/Page 2
Our main concern in chapter 17 is with sound waves in air, but sound can travel through
any gas, liquid, or solid (i.e. propane gas used in the demo)
Simplest sound waves are sinusoidal waves (have definite frequency, amplitude, and
wavelength)
Point source and spherical waves:
Sound is emitted from a point source S that emits sound waves in all directions. The
wavefronts and rays indicate the direction of travel and the spread of the sound
waves. Wavefronts are surfaces over which the oscillations due to the sound wave
have the same value;
For now we will focus on the idealized case of a sound that propagates (travels) in the
positive x-direction only.
Audible range:
humans: 20 to 20,000 Hz (top end gets lowers as one gets older)
(link to hearing test,
http://www.youtube.com/watch?v=h5l4Rt4Ol7M
1.5.kHz for me was highest)
PES 2130 Fall 2014, Spendier
Lecture 12/Page 3
2) Oscillating/periodic sound waves: traveling sound wave
Sound waves may also be described in terms of variations of pressure at various points.
In a sinusoidal sound wave in air, the pressure fluctuates above and below atmospheric
pressure in a sinusoidal variation with the same frequency as the motion of the air
particles.
So regions of high and low pressure correspond to places where the average displacement
of air is zero. Hence molecule displacement and pressure variation are π/2 rad (or 90o) out
of phase.
for sound waves: book uses s instead of y for displacement.
∆pm...pressure amplitude which is related to the displacement amplitude, sm , as follows:
∆pm = (vρω)sm
PES 2130 Fall 2014, Spendier
Lecture 12/Page 4
3) Speed of sound waves:
Remember the speed of wave on a string:
vstring 

tension in the string


linear mass density
In general (foe any traveling wave in any medium) we said:
For sound waves traveling in air and liquids:
1) inertia property: volume density of air ρ
2) restoring force: think of elastic property - for a string it was tension or stretchiness of
sting which we related to elastic potential energy (similar to stretched spring).
bulk modulus B: measures how easy it is to compress the molecules in a medium
B
p
V / V
[ Pa]
Here ∆V/V is the normalized change in volume produced by a change in pressure ∆p.
Think of air molecules in a piston (like we discussed in thermodynamics):
The air molecules randomly hitting the edges exert a force and a pressure on the surfaces.
If we reduce the volume by pushing the piston in, the normalized change in volume,
∆V/V<0, is negative and the pressure will go up, ∆p >0 (there is now a higher probability
for are molecules to hit the surfaces of the piston, creating a greater force and pressure).
If you plot ∆p as a function of ∆V/V, the Bulk modulus is the slope, and the smaller the
slope the easier it is to compress the air.
p  B  V / V 
Putting this together, we now have a expression for the speed of sound:
vsound , fluid 
B

. (in fluid or gas)
The speed of sound in a medium with bulk modulus B and density ρ.
PES 2130 Fall 2014, Spendier
Lecture 12/Page 5
Speed of sound in solid: When a longitudinal wave propagates in a solid rod or bar.
The rod will expand sideways (in contrast fluid in pipe has constant cross-section!)
Y
vsound , solid 
. (in solid)

Y... Young's modulus (is a measure of the stiffness of an elastic material)
Solids are more difficult to compress than gases or fluids. But they are also more dense
than gases. So, is the speed of sound greater or smaller in a solid compared to air?
v (air at 0oC) = 331 m/s
v (air at 20oC) = 343 m/s
v (steel) = 5941 m/s
v(granite) = 6000 m/s
(freezing point of water)
(room temperature)
So the speed of sound is generally larger in solids (B increases more than ρ)
Proof that ∆pm = (vρω)sm (see book, section before "interference")
4) Intensity and Sound Level
If you have ever tried to sleep while someone played loud music nearby, you are well
aware that there is more to sound than frequency, wavelength, and speed. There is also
intensity. The intensity I of a sound wave at a surface is the average rate per unit area at
which energy is transferred by the wave through or onto the surface. We can write this as
I
P
[W/m2]
A
where P is the time rate of energy transfer (the power) of the sound wave and A is the
area of the surface intercepting the sound. The intensity I is related to the displacement
amplitude sm of the sound wave
I
1
 v 2 sm2 [W/m2]
2
Variation of Intensity with Distance
How intensity varies with distance from a real sound source is often complex. Assume
that the sound source is a point source that emits the sound isotropically—that is, with
equal intensity in all directions. The wavefronts spreads (spherical waves) from such an
isotropic point source S and the intensity of sound from an isotropic point source
decreases with the square of the distance r from the source.
I
Ps
4 r 2
4 r 2 is the area of the sphere
Ps is the power of the source
PES 2130 Fall 2014, Spendier
Lecture 12/Page 6
The Decibel Scale
The displacement amplitude at the human ear ranges from about 10-5 m for the loudest
tolerable sound to about 10-11 m for the faintest detectable sound, a ratio of 106. From the
equation above, we see that the intensity of a sound varies as the square of its amplitude,
so the ratio of intensities at these two limits of the human auditory system is 1012.
Humans can hear over an enormous range of intensities.
We deal with such an enormous range of values by using logarithms. Thus, instead of
speaking of the intensity I of a sound wave, it is much more convenient to speak of its
sound level ß, defined as
 I 

 I0 
  (10dB )log 
dB is the abbreviation for decibel
I0 is a standard reference intensity (= 10-12 W/m2), chosen because it is near the lower
limit of the human range of hearing.
I is the intensity of the sound wave
For I = I0,
  (10dB)log 1  10(0)  0
So our standard reference level corresponds to zero decibels. Then ß increases by 10 dB
every time the sound intensity increases by an order of magnitude (a factor of 10). Thus,
ß = 40 corresponds to an intensity that is 104 times the standard reference level.
Note: Sound can cause the wall of a drinking glass to oscillate. If the sound produces a
standing wave of oscillations and if the intensity of the sound is large enough, the glass
will shatter.
PES 2130 Fall 2014, Spendier
Lecture 12/Page 7
Example:
Consider an idealized model with a bird (treated as a point source) emitting constant
sound power, with intensity inversely proportional to the square of the distance from the
bird. By how many decibels does the sound intensity drop when you move twice as far
away from the bird?