Mechanical Waves
A mechanical wave is a travelling disturbance in a
medium (like water, string, earth, Slinky, etc).
3: Mechanical Waves
(Chapter 16)
Move some part of the medium out of equilibrium,
and that motion travels (or propagates) from one
place in the medium to another.
• Since it took energy to disturb the medium from
Phys130, A01
Dr. Robert MacDonald
equlibrium, that energy propagates with the
disturbance. (Different parts of the medium are
moving as the wave moves; that’s kinetic energy.)
Waves on a string are a good basic model to use to
explore this.
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Waves: Space and Time
Snapshot at t=1 s of the below wave:
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Snapshot at time t = 0:
Delta y (mm)
2
Delta y (mm)
2 m/s
1
0
-1
-2
t=0
0
2
4
6
8
1
0
-1
-2
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x (m)
History of the string at x = 8 m:
t=1s
0
2
4
6
8
10
x (m)
History at x = 6 m, for a wave travelling right at 1 m/s:
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Delta y (mm)
Delta y (mm)
2
1
0
-1
x=8m
1
2
3
t (s)
4
0
-1
-2
-2
0
1
5
x=6m
0
1
2
3
t (s)
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Snapshot
at t=3 s of a wave travelling right at 1 m/s:
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Snapshot at t=0 s of the below wave:
2
Delta y
Delta y (mm)
1 m/s
1
0
-1
-2
t = 3s
0
2
4
6
8
1
0
-1
-2
10
x (m)
2
4
6
8
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History at x = 4 m, for a wave travelling left at 1 m/s:
History of the string at x = 2m:
2
1
0
-1
-2
0
x (m)
Delta y (mm)
Delta y (mm)
2
t=0s
0
1
2
3
4
2
4
6
2
Snapshot:
4
8
6
Delta x (cm)
1
2
3
4
5
Up to now we’ve been playing with pulses.
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Any wave with a repeating shape is a periodic wave. You
can have square waves, triangular waves, sinusoidal
waves, etc.
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Sinusoidal waves are the most common.
• Also the most important: it turns out you can
represent any wave as a combination of sinusoidal
waves. (This is called Fourier analysis.)
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Δx is displacement
from equlibrium
0
Periodic Waves
Wave pulse
0
-1
t (s)
Snapshot of a
Longitudinal Wave
0
0
-2
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t (s)
Equlibrium position
1
0.5
0
• Sinusoidal waves are generated by moving something
-0.5
-1
0
2
4
6
8
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in simple harmonic motion.
x (cm)
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1
v = 200 m/s
0.5
0
Snapshot at t=0
with wavelength
λ halved:
-0.5
-1
0
1
2
3
4
Delta y(x, t=0) (mm)
Delta y(x, t=0) (mm)
Snapshot
at t=0
1
0.5
0
-0.5
-1
0
5
1
2
Snapshot at t=0
with frequency f
halved but speed
unchanged:
1
0.5
0
-0.5
-1
0
1
2
3
4
Delta y(x, t=0) (mm)
Delta y(x, t=T/4) (mm)
Snapshot
at t=T/4
Delta y(x, t=T/2) (mm)
4
5
3
4
5
1
0.5
0
-0.5
-1
0
5
1
2
x (m)
x (m)
Snapshot
at t=T/2
3
x (m)
x (m)
1
0.5
0
-0.5
-1
0
1
2
3
4
5
x (m)
Longitudinal
Waves
Rarefaction
(lower density)
Compression
(higher density)
Speaker
Periodic:
Rarefaction
(lower density)
Compression
(higher density)
Rarefaction
(lower density)
Compression
(higher density)
• Waves transport energy, not matter.
wavelength (λ)
0
Pulse:
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A wave is a disturbance from equlibrium, propagating
through some medium (like a string).
It takes energy to disturb a particle from equlibrium,
so energy must be travelling through the medium
causing the disturbance.
v
Equlibrium position
What Waves Are
We’ll be using waves on a string as our basic model.
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“Waves on a String” Simulation:
http://phet.colorado.edu/simulations/sims.php?sim=Wave_on_a_String
Wave pulse
0
2
4
6
8
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Wave Speed
Particle on a
String
In a transverse sinusoidal wave,
each bit of string is moving
vertically in simple harmonic
motion.
Particles in a fluid move back
and forth in SHM as well during
a longitudinal sinusoidal wave.
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Fig. 16.4
Example: Tsunami
An earthquake off the coast of Sumatra on
26 December, 2006, sent a tsunami smashing into
southeast Asia. Satellites measured the wavelength to
be about 800 km (!), and a period between waves of
one hour.
• What was the wave speed, in km/h and m/s?
By the time a particle has
completed one cycle of SHM
(e.g. moved down and back up),
the wave has moved forward by
one wavelength (λ).
The time it takes for the particle
to do this is one period (T).
So, since the wave speed v is
constant, v = λ/T or v = λf.
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Fig. 16.4
Example: Ultrasound
Ultrasound — sound with frequency too high for
humans to hear (above about 20 000 Hz) — is a useful
tool for medical imaging. Send the sound waves into the
body, and listen to the echoes off of various tissues and
bones etc. The speed of sound in body tissue is typically
about 1500 m/s.
• To get a clear image, considering the typical size of
what the doctors are looking at, the wavelength of
the sound should be about 1 mm or less. What is
the minimum frequency they should use?
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Describing wave shape
We’ve already described the way a particle in SHM
moves over time (history graph!) back in the previous
chapter:
!
y(t) = A cos(ωt + ϕ).
• Each point on the string (or whatever’s waving) is
moving up and down with the same period, but
with different phase. One point is at a peak,
another at a trough, another moving through the
equilibrium position, etc.
• In other words, each point on the string has a
What about the shape of a wave in space at a given
moment (snapshot graph!)?
It’s still sinusoidal, so we can use cosine (or sine).
In time (history graph), we had a phase that looks like
“ωt + ϕ”. A phase like that gave us a nice cosine curve
with the right period, and it let us choose t = 0 to be at
any point in the cycle.
Let’s try something similar with position. We can’t use
ω since the units are wrong (time, not space), so we
have to come up with something else...
different initial phase ϕ, so ϕ depends on x. We can
write this as “ϕ(x)”.
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So we need to translate a position in space (x) into a
phase angle so we can use cosine.
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Waves in Time and Space
If x = λ is one wavelength from the origin, then x/λ is the
fraction of a wavelength we are away from the origin.
Now we’d like to put it all together, to get a
wavefunction that describes the wave’s behaviour in
time and in space.
There are 2π radians in one cycle. So 2πx/λ should
give the correct phase!
Each bead or bit of string is moving in simple harmonic
motion, with a different phase. The phase, in other words,
depends on x. So we can describe this by
! y(t) = A cos(ωt + ϕ(x)).
For convenience, define the wavenumber:
This is not the
spring constant!
y(x) = A cos(kx + ϕ0)
2π ω
Also useful: !v = λf =
, so v = ω/k.
k 2π
So in space, then:
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If you move over a distance x, the phase will be different by
kx = 2πx/λ. Exactly what phase you end up with depends
on what phase you had at the origin, just like with time, so
we need a phase constant ϕ(x=0) = ϕ0. Then
! ϕ(x) = 2πx/λ + ϕ0.
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Fig. 16.4
But what about the sign? Should we use
y(x, t) = A cos(kx – ωt) or
y(x, t) = A cos(kx + ωt)? It’s not obvious!
Going the Other Way
So far we’ve been describing waves moving to the
right (positive x direction). What about waves moving
to the left?
Consider the first point that starts on axis
at t=0 in the diagram to the right. Its phase
at this time is 3π/2.
The behaviour of a bit of string as time proceeds is
now just the opposite of what it was before. So we
just need to flip the sign on the time-dependent term
in our phase — i.e. replace –ωt with +ωt.
Increasing the phase would move this bit of
string up at first. But instead, as the wave
moves to the right this bit goes down!
In order to describe this, we need to use
–ωt; this way the phase decreases as time
increases.
cos(theta)
So a sinusoidal wave moving to the right can
be described by:
y(x, t) = A cos(kx – ωt + ϕ0)
!
So depending on direction, our wavefunction is:
graph of
cos(θ)
1
0.5
0
1
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! y(x, t) = A cos(kx + ωt + ϕ0)
Sinusoidal wave moving
in –x direction.
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3
4
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Phase (theta), radians
increasing phase !
More on Phase
We’ve put back the phase constant ϕ0:
!
y(x, t) = A cos(kx – ωt + ϕ0)
ϕ0 represents the phase at (x, t) = (0, 0).
The value (kx – ωt + ϕ0) gives the phase at any point in
space and time.
•
For a crest (y = +A), the phase can be 0, 2π, 4π,
–2π. –4π, etc.
•
For a trough (y = –A), the phase can be π, 3π, 5π,
–π, –3π, etc.
v = ω/k (or λf) is often called the “phase velocity” vp.
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Sinusoidal wave moving
in +x direction.
0
-0.5
-1
This is called a wavefunction.
! y(x, t) = A cos(kx – ωt + ϕ0)
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Example: Birds on a String
Two tiny birds are sitting 3.00 m apart on a long, heavy
rope. Some joker comes along and starts shaking one
end of the rope in SHM, with a frequency of 2.00 Hz
and an amplitude of 0.075 m. The speed of the
resulting wave is 12.0 m/s. At time t=0 the person’s
hand is at the top of its motion (maximum positive
displacement).
• What are the amplitude, angular frequency, period,
wavelength, and wave number of the wave?
• What is the difference in phase between the two
birds?
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So now you know:
• what waves are.
• what actually travels along a wave (energy!).
• how a particle moves as a wave passes through.
• how the frequency, wavelength, and speed of a wave
are related.
• how to describe a wave mathematically.
Aside: Partial Derivatives
A partial derivative is the type of derivative you
need to use when you have a function of more than
one variable, such as the wavefunction y(x, t).
It works the same as a regular derivative except you
replace d with ∂ (a sort of curly “d”).
It’s basically a derivative that winks at you ;-) and says,
“We both know this is a function of x and t, but I’ll
just pretend it’s only a function of one of them for
now.” You take the derivative with respect to, say, x,
treating t as just another constant (or vice versa).
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The Wave Equation
Pretty much all waves follow something called the
wave equation. (Don’t confuse this with the wave
function, which is y(x,t) and describes a specific wave!)
Vertical displacement
of the string at x:
Vertical velocity
of the string at x:
The wave equation looks like this:
(You don’t need this on
your formula sheet.)
It’s a relationship between the curvature of the wave (at
some location, at some time) and the acceleration of
the particle in the wave at that same place and time.
(We’ll be using it to find v (the wave speed) for some
specific types of waves.)
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These are the same SHM equations of motion we
already know, but the initial phase (the phase when t=0)
at position x is kx + ϕ0. (Compare with A cos(ωt + ϕ).)
We can also study the shape of the string (or whatever’s
waving) at some specific snapshot in time, by looking at
how the displacement y varies with x...
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The Wave Equation
Slope of the string:
Solve both the particle acceleration and the string
curvature for y and equate them:
Curvature of the string:
Now this is interesting... compare to the accelleration:
Wave speed, not
particle speed!
Remember that wave speed v = ω/k, so:
The Wave
Equation
Both are a constant times y. And they’re similar
constants: (2π/λ)2 vs (2π/T)2.
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“Applying” the Wave Equation
To get a wave equation for a particular type of wave,
you need these ingredients:
• A restoring force (F ).
• Newton’s second law (F = ma).
• Lots of linearization and simplification (calculus!).
net
net
Our purpose: once we’re done, we’ll be able to read v
(the wave speed) directly from the resulting wave
equation.
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