Vibrations and waves
Physics 123, Spring 2007
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Lecture III
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Lecture III: Vibrations and Waves
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SHO: resonance
Waves
Transverse and longitudinal waves
Reflection
Interference
Standing waves
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Waves
Matter
• Propagating oscillation =
wave.
• Waves transport energy
and information, but do
not transport matter.
• Examples:
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Wave
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Ocean waves
Sound
Light
Radio waves
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Waves
• Wavelength – l
• Period T
• Frequency f=1/T
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• Wave velocity:
v=l/T=lf
The only equation that you need
to remember about waves.
• Wave velocity is NOT the
same as particle velocity of
the medium
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Transverse and longitudinal
waves
• In transverse wave the velocity of particles of
the medium is perpendicular to the velocity of
wave.
Matter
Wave
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Transverse and longitudinal
waves
• In longitudinal wave the velocity of particles of
the medium is parallel (or anti-parallel) to the
wave velocity.
Wave
Matter
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Description of waves
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w=2pf – cyclic frequency, k=2p/l –wave vector
D=D0sin(kx-wt+d), d-phase at t=0, x=0
Riding the wave kx-wt+d=const
kx-wt=c
x=c/k+(w/k)t = x0+vt
Thus, wave velocity v=w/k=2pf/ (2p/l)=fl = l/T
D=D0sin(kx-wt) – wave is moving in +x direction
D=D0sin(kx+wt) – wave is moving in -x direction
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Average intensity
• Displacement D follows harmonic oscillation:
D = D0 sin( t )
• Intensity (brightness for light) I is proportional to
electric field squared
I  D 2  I = I 0 sin 2 (t )
• Average over time (one period of oscillation) I:
T
T
1
1
2
2
I = I 0  sin (t )dt = I 0
sin (t )dt =

T 0
T 0
1
= I0
2p
2p
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2p
I0
1 1
2
0 sin xdx = I 0 2p 2 0 (1 - cos 2 x)dx = 2
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Energy transported by waves
• Intensity of oscillation I
(energy per unit area/ per sec)
is proportional to amplitude
squared D2
• 3D wave (from energy
conservation):
D12 4pr12= D22 4pr22
D1/D2=r2/r1
• Amplitude of the wave is
inversely proportional to the
distance to the source:
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D
r
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Interference of waves
• When two or more waves pass through the same
region of space, we say that they interfere.
• Principle of superposition (fancy word for sum of
waves): the resultant displacement is the
algebraic sum of individual displacements created
by these waves.
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Constructive and destructive
interference
in phase
Constructive
out of phase
Destructive
not in phase
Partially destructive
A
2A
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<A
0
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Adding waves
• Two waves observed at a certain location, can set it to be
x=0: D1=D10sin(wt), D2=D20sin(wt+d)
• Suppose for simplicity D10=D20
• (principle of superposition):
D1 = D0 sin( wt )
D2 = D0 sin( wt  d )
D1  D2 = D0 (sin(( wt  d / 2) - d / 2)  sin(( wt  d / 2)  d / 2) =
= 2 D0 sin( wt  d / 2) cos(d / 2)
• Amplitude of oscillation 2D0cos(d/2) is determined by
the relative phase shift d
• Intensity of the sum is proportional to cos2(d/2)
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Reflection of a transverse wave
pulse
•Reflection from fixed
end –inverted pulse
d=p

•Reflection from loose
end – the pulse is not
inverted
d=0

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Standing waves
• Interference of a
wave with its
reflection creates a
standing wave.
• Only standing waves
corresponding to
resonant frequencies
(e.g. nodes at fixed
ends) persist for long.
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Standing waves
• Add two waves traveling in
opposite directions:
• D1 =Dsin(kx-wt)
• D2 =Dsin(kx+wt)
D=D1+D2=2Dsin(kx)cos(wt)
• Boundary condition
• Sin(kL)=0  kL=pn
• K=pn/L; l=2L/n
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Standing waves
• First harmonic or
fundamental frequency:
– L=l1 /2
– f1=v/l1
l1=2L
f1=v /(2L)
• Second harmonic:
– L=l2
– f2=v /l2
– f2=v /L=2f1
l2=L
• N-th harmonic:
– L=nln /2
– fn=v /ln
–
nv
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fn =
2L
ln=2L/n
= nf1
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