Chapter
p
15
Mechanical Waves
PowerPoint® Lectures for
University Physics, Thirteenth Edition
– Hugh
g D. Young
g and Roger
g A. Freedman
Lectures by Wayne Anderson
Copyright © 2012 Pearson Education Inc.
Goals for Chapter 15
• To study the properties and varieties of mechanical waves
• To relate the speed, frequency, and wavelength of periodic waves
• To interpret periodic waves mathematically
• To calculate the speed
p
of a wave on a stringg
• To calculate the energy of mechanical waves
• To understand
d
d the
h interference
i
f
off mechanical
h i l waves
• To analyze standing waves on a string
• To investigate the sound produced by stringed instruments
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Types of mechanical waves
• A mechanical wave is a disturbance traveling through a medium.
medium
• Figure 15.1 below illustrates transverse waves and longitudinal
waves.
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The Wave Equation
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Solutions to the wave equation
November
13 2012
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Physics 208
6
Periodic transverse waves
• For the transverse waves shown
here in Figures 15.3 and 15.4, the
particles move up and down, but
the wave moves to the right.
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Periodic longitudinal waves
• For the longitudinal waves shown
here in Figures 15.6 and 15.7, the
particles oscillate back and forth
along the same direction that the
wave moves.
• Follow Example
p 15.1.
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The solutions to the
Wave Equation…
what do they look
like???
November
13 2012
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Physics 208
9
November
13 2012
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Physics 208
10
November
13 2012
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Physics 208
11
The Wave Equation for y(x,t)
∂ y μ ∂ y
=
2
2
∂x
T ∂t
the solutions for y(x,
y(x t) are
f(t-x/v) and g(t + x/v)
2
with
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2
v=
T
μ
Choosing our favorite solution
y ( x , t ) = A cos( kx − ω t )
with
x
t − = ω t − kx
v
where
k=
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2π
λ
, ω = 2πf andd v = λ f =
ω
k
Traveling waves
March 1,
2012
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Physics 221
14
Particle velocity and acceleration
of the medium
y ( x , t ) = A cos( kx − ω t )
then
∂y ( x , t )
= ω A sin( kx − ω t )
v y ( x, t ) =
∂t
and
∂ y ( x, t )
2
= −ω A cos( kx − ω t )
a y ( x, t ) =
2
∂t
2
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Energy carried in a wave
∂y ( x , t )
Fy = − F
∂x
with
P ( x, t ) = F y ( x, t )v y ( x, t ) =
∂y ( x , t ) ∂y ( x , t )
−F
∂x
∂t
March 1,
2012
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Physics 221
16
Average Power in a sine wave on a string
P(x,t) = μF ω A sin ( kx − ωt )
2
2
2
with
Pave
1
2 2
=
μF ω A
2
Note: One half of the maximum power transmitted in the
wave Also note the frequency and amplitude
wave.
dependence of these mechanical waves.
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Wave Intensity
Intensity =
the time averaged power per unit area
P
I = ave
4πr 2
units are watts/meter 2
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Reflection from a fixed end and
a free end.
Note: The phase changes are
different for the two reflected
waves.
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Superposition of mechanical waves
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The mathematics of wave superposition
ytotal
t t l(x,t) = y1(x,t) + y2(x,t)
with
ytotal also a solution to the wave equation
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Standing Waves on a string
y1(x,t) = − A cos(kx + ωt ) and
y2(x,t) = A cos(kx − ωt )
ytotal
then
= A(− cos((kx + ωt ) + cos((kx − ωt ))
using cos(a ± b) = cos a cos b m sin a sin b
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Standing waves on a string
• Waves traveling in opposite directions on a taut string
interfere with each other.
• Th
The result
lt is
i a standing
t di wave pattern
tt
that
th t does
d
nott move
on the string.
• Destructive interference occurs where the wave
displacements cancel, and constructive interference
occurs where
h th
the displacements
di l
t add.
dd
• At the nodes no motion occurs, and at the antinodes the
amplitude of the motion is greatest.
15.23
23 on the next slide shows photographs of
• Figure 15
several standing wave patterns.
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Photos of standing waves on a string
• Some time exposures
p
of standingg waves on a stretched string.
g
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The formation of a standing wave
• In Figure 15.24,
15 24 a
wave to the left
combines with a
wave to the right
to form a standingg
wave.
• R
Refer
f to
t ProblemP bl
Solving Strategy
15 2 and follow
15.2
Example 15.6.
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Normal modes of a string
• For a taut string fixed at
both ends, the possible
wavelengths are λn = 2L/n
and the possible frequencies
are fn = n v/2L = nf1, where
n = 1,
1 2,
2 3,
3 …
• f1 is the fundamental
f
frequency,
f2 is
i the
th secondd
harmonic (first overtone), f3
is the third harmonic
(second overtone), etc.
• Figure 15
15.26
26 illustrates the
first four harmonics.
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Standing waves (continued)
y total = A( − cos(( kx
k + ω t ) + cos(( kx
k − ω t ))
y total = ( 2 A sin kx ) sin ω t
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Location of the nodes for a standing wave on a
string.
ytotal = ( 2 A sin kx ) sin ωt
for a string fixed at x = 0,
the nodes will be located at points where
kx = mπ , with x = 0, λ / 2, λ , 3λ / 2, 2λ ...
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Normal Modes of a String
The string is rigidly
held at both ends, so
we have two sets of
b
boundary
d
conditions
diti
to be satisfied.
L = n λ/2
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Normal Modes (continued)
λn = 2 L / n where n = 1,2,3,4..
f1 = v / 2 L the fundamental frequency
f n = n(v / 2 L) give the harmonic frequencies
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