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Natural Sciences I
lecture 10: Waves and Vibrations
ELASTIC MATERIALS
Our initial discussion of waves and vibrations will address the behavior
of elastic materials, so we need a clear definition with which to proceed. An
elastic material is one that restores itself to its original dimensions and
shape after being deformed. The most familiar example of an elastic
material is a spring, which can be deformed extensively and still recover its
shape. Many materials and objects not specifically designed for use as
springs nevertheless exhibit elastic behavior for limited amounts of
deformation. All materials have what is called an elastic limit, which is the
maximum deformation from which the material can recover its original state.
For a given spring, there is a specific relationship between the amount of
force applied in stretching or compressing it and the extent of deformation: the
greater the force, the greater the deformation. For a given spring, there is
also a characteristic relationship between the magnitude of the internal
restoring force and the extent of deformation.
VIBRATIONS
A vibration is a repeating back-and-forth motion, as would be exhibited
by the spring below (this is a simple harmonic oscillator)...
equilibrium
position
stretched
m
relaxed
compressed
frictionless
surface
m
m
max. velocity
start the system
oscillating with an
external force
external force
restoring force
FR
m
"overshoot"
(note: FR
displacement)
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The periodic, or oscillatory, vibration of the mass in the previous figure
is called simple harmonic motion, which we can define as vibratory
motion that occurs whenever there is a restoring force opposing (and
proportional to) a displacement. This kind of motion is very commonplace in
natural systems:
spring & mass
bent bar (diving board)
taut string (musical instrument)
air in hollow tube (musical instrument)
molecules and atoms
electromagnetic waves
electrons in atoms
We need to define some terms before we go further (more jargon!)...
AMPLITUDE
maximum displacement from the equilibrium position
CYCLE
one complete vibration (oscillation)
PERIOD (T)
time to complete one cycle
FREQUENCY (f)
number of cycles per unit time [units = Hertz (Hz)]
1 Hz = 1 cycle/s
T = 1/f; f = 1/T; T
f=1
Example calculation: A vibrating system has a period of 0.001 seconds.
What is the frequency of the system in Hz?
T = 0.001 s
f=?
f = 1/T
= 1 / 0.001 s
= 1000 s
-1
= 1000 Hz
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GRAPH of simple harmonic motion...
amplitude
cycle
x
paper
equilibrium position
x=0
Fnet = 0
a=0
v = vmax
max. displacement
v=0
How are restoring force (displacement), velocity, and acceleration
related to time?
displacement
(restoring force)
zero
velocity
zero
acceleration
zero
time
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WAVES
As noted previously, a vibration is characterized by repeating backand-forth motion of a system that exhibits restoring forces proportional to
displacement. If a vibration occurs within another medium (say, air or
water), it can create a WAVE. Waves can also be created by a one-time
disturbance called a pulse. For now, we'll define a wave as a propagating
disturbance of a medium produced by a vibration or a pulse.
There are two kinds of waves:
Transverse Waves in which the motion of the particles that make up
the medium is perpendicular to the direction of wave propagation.
Transverse waves are sometimes called shear waves, and they can
propagate only in materials that have shear strength – that is, some
attachment between adjacent particles or molecules (most liquids have
little or no shear strength, and gases definitely have none).
A brief digression on shear strength is needed...
(this experiment requires a molecular hammer; do
not attempt without supervision)
SOLID: molecules are
strongly bonded to one
another, so a "transverse"
disturbance is passed
"down the line".
disturbance
propagated
solids have shear strength
LIQUID (or gas): if you
could do the same experiment with a liquid or gas,
you would simply shear
off a few molecules:
nothing would be "felt" by
those not directly struck
by the hammer.
undisturbed
liquids have no
shear strength
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Some familiar transverse
waves are those that can be
propagated along a rope, rug, or
slinky by shaking one end
propagation
direction
The particles making up the
propagation medium do not
approach one another or separate as the wave passes – they
move in unison perpendicular to
the direction in which the wave
propagates.
Earthquakes "s" waves are
transverse waves. They do not
pass through the Earth's molten
metallic core, and they are
attenuated in squishy regions of
the Earth.
LONGITUDINAL WAVES are disturbances that cause the particles of
the propagation medium to alternately approach and separate from one
another as the wave passes. The motion of the particles is in the direction
of wave propagation, as in the examples below. Longitudinal waves are
also referred to as "compressional" waves, and they can propagate
through both liquids and gases as well as solids. In seismology they are
referred to as "p" waves. They travel through all parts of the Earth and
propagate a little faster than transverse waves.
SLINKY (spring)
AIR or WATER
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The diagrams at the bottom of page 5 depict an isolated or "solitary"
wave that would be caused by a single pulse. A vibrating disturbance in air
(for example) produces alternating regions where the molecules are more
and less densely packed – "condensations" and "rarefactions":
These radiate outward from the
source in all directions...
condensations
(increased pressure)
rarefactions
(decreased pressure)
Remember that the pressure
of the gas is proportional to the
number of molecules per unit
volume, so the condensations
are local high pressure regions
and the rarefactions are local
low-pressure regions.
Hearing sound waves in the air
Your eardrum – like any microphone – is really just a pressure sensor:
it responds to pressure differences as condensations and rarefactions pass
by. The human ear responds to frequencies between ~20 Hz and 20,000 Hz.
infrasonic
< 20 Hz
ultrasonic
> 20,000 Hz
The Wave Equation
v = lf = l/T
v = wave speed
f = frequency
T = period
l = wavelength
Note: The wavelength is the distance between
wave crests or troughs. These crests and
troughs can be physical displacements
(transverse waves) or maximum and minimum
compression regions
l
l
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Using the Wave Equation: an example...
A sound wave with a frequency of 360 Hz has a wavelength of 0.92 m.
At what speed does this sound wave propagate?
f = 360 Hz
l = 0.92 m
v = ?
v =
=
=
=
lf
(0.92 m) (360 s-1)
(0.92) (360) m/s
331 m/s
Variables that affect sound speeds in air...
frequency of sound (very small effect)
temperature
moisture content of air
frequency (Hz)
speed (m/s)
20
8000
o
(dry air at 20 C)
343.48
343.57
pressure (very small between 1 & 2 atm)
moisture
temperature
o
344.4
20 C
100 Hz
sound speed (m/s)
sound speed (m/s)
344.8
344.0
343.6
0
380
dry air
340
300
o
0C
260
20 40 60 80 100
relative humidity (%)
-100 -60 -20 20 60 100
o
temperature ( C)
Here's an equation for temperature dependence of sound speed (dry air):
vT (m/s) = v0 +
P
0.60 m/s
TP
o
C
v0 ~ 331 m/s (dry air)
Note: This is a linear equation – so it's an approximation to the curve in the
right-hand graph above.
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In general, the speed of sound in a particular material depends upon
two factors: the inertia (mass) of the molecules and the strength of the
bonds between them. This means that the speed of sound is highest in
"low-mass" materials (e.g. hydrogen or helium gas) and in very rigid
materials...
material or
medium
sound speed
(m/s)
dry air (0oC)
331
helium (0oC)
965
hydrogen (0oC)
1,284
water (25oC)
1,497
lead metal
1,960
glass
5,100
steel
5,940
aluminum
6,420
beryllium* metal
* atomic no. 4
12,890
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REFRACTION and REFLECTION: Behavior of waves at interfaces and in
changing media
Most sound waves originate from a localized source if not from an actual
point. This means the waves move outward from the source in spherical
"fronts", as noted in the last lecture. If the waves travel far enough, however,
the spherical fronts become essentially planar...
"planar" wave fronts
One of the questions we're asking today is What happens to these wave
fronts when they encounter another medium? There are several possibilities.
If a wave strikes a boundary that is parallel to the wave front, it may be
absorbed
transmitted
reflected
or some combination of the three.
If a wave strikes a boundary at an oblique angle, we can add refraction to
the list of possibilities.
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Absorption and reflection are familiar phenomena, but refraction is more
challenging to our intuition because it involves a change in direction at a
boundary or interface ("bending" of waves):
1
w
e
av
v=lf
s
nt
o
fr
faster speed
slower speed
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3
w
e
av
The frequency of the
propagating waves must
remain the same as they
pass from one medium
to another. This means
that if v is slower in the
green medium, then l
must be shorter
fro
s
nt
faster v
slower v
w
lf
ls
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simple representations
of wave refraction
RESULT
s
ont
r
f
ave
2
slower v
faster v
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SUMMARY of wave behavior at interfaces
transmission
(without refraction)
reflection
refraction
wave fronts
interface
1
2
a
a
b
b
a=b
v
sin a
= 1
v2
sin b
Refraction examples
Loudness of aircraft to ground observer
warm (fast)
cool (slow)
wave fronts
propagation
warm (fast)
cool (slow)
Seismology
wave fronts
propagation
paths
Earthquake
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Reflection examples
ECHO in a large room
d = ½ t vs
vs = 343 m/s
at room temperature
wall or other
smooth surface
d
t = time elapsed
between clap and
hearing the echo
Sonar
d
d = ½ t vsw
vsw = 1531 m/s
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INTERFERENCE
In contrast to particles, waves can coexist in the same space with
other waves. When this happens, the waves interact with one another in
various ways: the phenomena are referred to as interference. Waves that
are in phase experience constructive interference; waves that are out of
phase experience destructive interference...
First, some examples using waves of the same wavelength (l)...
The red and blue waves at the right
are of different amplitude, but they are
exactly in phase (all the peaks and
valleys line up). The interaction
between them would be constructive,
producing the purple wave.
a
b
a+b
CONSTRUCTIVE INTERFERENCE
a
b
a+b
The red and blue waves at the left
are of the same amplitude and
exactly out of phase – they cancel
each other completely.
DESTRUCTIVE INTERFERENCE
a
b
a+b
DESTRUCTIVE INTERFERENCE
The red and blue waves at the left
are exactly out of phase (as in the
previous example, but they don't
completely cancel each other
because b has a larger amplitude.
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If two sources produce waves of differing wavelength, more complex
possibilities arise...
separate a and b waves of slightly different frequency
a + b interference gives a wave whose amplitude varies periodically
This phenomenon is referred to as a "beat". Examples include:
aircraft with twin engines running at slightly different speeds
two snow tires on a vehicle speeding down the highway
twin diesel locomotives
Note: We have been illustrating interference phenomena using transverse
waves (because they're easier to draw), but all the same effects apply to
longitudinal waves – a fact that is probably obvious from the sound wave
examples above.
a
b
a+b
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ENERGY and SOUND
All waves involve the transport of energy in the form of motion or
compression of the constituent particles of the wave-transmitting medium. In
the case of sounds waves, we measure energy-moving effectiveness in terms
of the intensity of the wave:
power
area
intensity =
I =
P
A
A
2
units = watts/m
The human ear is
extremely sensitive
– it is capable of
detecting sounds
with intensities as
-12
2
low as 10 W/m .
Discomfort comes
2
at ~ 1 W/m .
Remember that power is work per unit time (W / t). It should be intuitively
reasonable that the power of the sound wave – that is, the amount of work
it can do in a given time – depends upon the amplitude and frequency of
the wave...
high pressure
low pressure
l
The amplitude reflects the
"peak-to-trough" variation
in air pressure, which
determines how much
work a single pulse can
do (or how much kinetic
energy it can impart)
when it hits the wall. The
wavelength or frequency
determines how many
pulses hit the wall in a
given amount of time.
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More on sound detection by the human ear...
Most of us are familiar with the term decibel in connection with
sound intensity. The decibel scale is a relative scale – it is the ratio of
-12
2
sound intensity to the threshold of hearing (10 W/m ). It is also a log
scale:
example
barely audible
decibels
intensity
2
(W/m )
0
1E-12
whisper
20
1E-10
heavy traffic
70
1E-5
pneumatic drill
95
3E-3
120
1
jet aircraft
RESONANCE and NATURAL FREQUENCY
All objects possess what is called a natural frequency of vibration,
which is determined by the material and the shape of the object. You
know, for example, that water glasses of different sizes and shapes "ring"
at different frequencies when struck. If sound waves impinging on an
object coincide with the natural frequency of that object, it will begin
vibrating "spontaneously". The phenomenon is called resonance.
(For this experiment
to work, the natural
frequencies of the
two tuning forks must
be exactly the same)
strike this one...
...this one begins
to vibrate
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Vibrating strings are subject to the principles of natural frequency and
resonance, plus some additional constraints:
incoming wave
fixed end
taut string
(amplitude exaggerated)
reflected wave
fixed end
If both ends of the string are fixed, a "standing wave" is produced...
antinodes
fixed
fixed
node
node
node
L
Waves traveling down the string are reflected back at the fixed ends
Whenever l = 2L/n, the reflected waves will constructively interfere with
the incoming waves to amplify the vibration and create a standing wave
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L
l1
fundamental
frequency
= 2L
st
(1 harmonic)
l2 = L
l3 = 2L/3 overtones
l4 = L/2
Overtones occur when l = 2L / n (n is an integer)
Which standing wave(s) is sustained depends upon the natural
(resonant) frequency of the string – which depends on mass,
tension and length.
NOTE: The vibrational modes of the string are quantized!