PHYS151
Lecture 17
Ch 17 Waves - I
Eunil Won
Korea University
Fundamentals of Physics by Eunil Won, Korea University
Types of Waves
Mechanical waves:
water waves, sound waves, seismic (지진) waves
Electromagnetic waves:
visible light, radio and television waves,
microwaves, x rays : all electromagnetic waves
travel through a vacuum at the same speed c
c = 299 792 458 m/s
Matter waves:
matter can be considered as wave in
microscopic level : electrons, protons
Fundamentals of Physics by Eunil Won, Korea University
Transverse and longitudinal Waves
A single pulse can be sent with a certain velocity
If you move your hand up and down in simple harmonic
motion, a continuous wave travels along the string
Transverse wave: the displacement of the oscillating
element is perpendicular to the direction of travel of
the wave
Longitudinal wave: the displacement of the oscillating
element is parallel to the direction of travel of the
wave
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Sand Scorpion (전갈)
The sand scorpion uses waves of both transverse and longitudinal motion to
locate its prey
When a beetle (딱정벌레) disturbs the sand, it sends
pulses along the sand’s surface
longitudinal pulse: vl = 150 m/s
transverse pulse: vt = 50 m/s
The scorpion, with its eight legs, intercepts
1) the longitudinal pulse for direction
2) Δt between two pulses for distance d
d
d
∆t =
−
vt
vl
d = (75 m/s)∆t
For example, if Δt = 4.0 ms, then d = 30 cm, which
gives the scorpion a perfect fix on the beetle
Fundamentals of Physics by Eunil Won, Korea University
Wavelength and Frequency
At time t, the displacement y of the element located at position x is given by
y(x, t) = ym sin (kx − ωt)
amplitude: the magnitude of the
maximum displacement of the element
wavelength : distance λ between
repetitions of the shape of the wave
at time t=0,
y(x, 0) = ym sin kx
displacement y is the same at x=x1 and x=x1 + λ
ym sin kx1
= ym sin k(x1 + λ)
= ym sin (kx1 + kλ)
A sine function repeats itself when it
angle is increased by 2π rad:
Fundamentals of Physics by Eunil Won, Korea University
2π
k=
λ
Wavelength and Frequency
Below shows a graph of the displacement y versus time t at a certain position (x=0)
y(0, t) = ym sin (−ωt)
= −ym sin ωt
period of oscillation T: time that any string element
takes to move through one full oscillation
so,
−ym sin ωt1
= −ym sin ω(t1 + T )
= −ym sin (ωt1 + ωT )
angular frequency:
2π
ω=
T
frequency:
ω
1
=
f=
T
2π
Like the frequency of simple harmonic motion in the previous chapter, this
frequency is a number of oscillations per unit time:
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The Speed of a Traveling Wave
Figure shows two snapshots of the wave,
taken a small time interval Δt apart. The ratio
Δx/Δt is the wave speed. How can we find it?
As the wave moves, each point in the string retains its displacement y:
kx − ωt = constant
ω
dx
=v=
dt
k
λ
ω
= λf
v= =
k
T
A wave traveling in the negative x direction can be found by replacing t with -t:
y(x, t) = ym sin (kx + ωt)
so the resulting speed becomes:
Fundamentals of Physics by Eunil Won, Korea University
ω
dx
=−
dt
k
Wave Speed on a Stretched String
The speed of a wave is set by the properties of the medium
consider a small string element:
Δl : length
R : radius (of the circle formed)
τ : force equal to the tension in the string
the vertical component of the force forms a radial
restoring force:
∆l
F = 2(τ sin θ) ≈ τ (2θ) = τ
R
the mass of the string element: ( μ : string’s linear density)
∆m = µ∆l
centripetal acceleration:
v2
a=
R
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F = ma gives
v2
τ ∆l
= (µ∆l)
R
R
v=
!
τ
µ
Energy and Power of a Traveling Wave
Kinetic energy: an element of string oscillates
transversely in simple harmonic motion (providing
kinetic energy)
Elastic potential energy: as a string element
oscillates, elastic potential energy changes (dx
changes)
Energy transport: as the wave moves, energy is
transfered (wave transports the energy)
kinetic energy dK associated with a string element dm
(u: transverse speed):
u: transverse speed is
dm = μ dx :
rate of the kinetic
energy transferred :
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1
dK = dm u2
2
∂y
= −ωym cos (kx − ωt)
u=
∂t
1
dK = (µ dx)(−ωym )2 cos2 (kx − ωt)
2
1
dK
2
= µvω 2 ym
cos2 (kx − ωt)
dt
2
Energy and Power of a Traveling Wave
The average rate at which kinetic energy is
transported:
!
dK
dt
"
1
2
µvω 2 ym
[cos2 (kx − ωt)]avg
2
1
2
µvω 2 ym
4
=
avg
=
1
2π
!
2π
cos2 x dx =
0
=
1
2π
1
2
!
0
2π
"
1
1
cos 2x +
2
2
the average kinetic energy and the average potential energy are equal:
Pavg = 2
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!
dK
dt
"
avg
or
Pavg
1
2
= µvω 2 ym
2
#
dx
The Principle of Superposition for Waves
Two waves travel simultaneously along the same string: the
displacement of the string when two waves overlap is
y ! (x, t) = y1 (x, t) + y2 (x, t)
: principle of superposition
Overlapping waves algebraically add to produce a
resultant wave (or net wave)
Overlapping waves do not in any way alter the travel of
each other
Fundamentals of Physics by Eunil Won, Korea University
Interference of Waves
Two sinusoidal waves with same wavelength and amplitude: principle of superposition applies
If two waves are in phase (peaks and valleys are exactly aligned), the
net displacement is doubled
If two waves are out of phase (peaks and valleys are exactly
opposite), the net displacement is zero
Let us assume:
y1 (x, t) = ym sin (kx − ωt)
y2 (x, t) = ym sin (kx − ωt + φ)
The resulting displacement is then:
y ! (x, t)
= y1 (x, t) + y2 (x, t)
= ym sin (kx − ωt) + ym sin (kx − ωt + φ)
since
1
1
sin α + sin β = 2 sin (α + β) cos (α − β)
2
2
1
2
It becomes y ! (x, t) = [2ym cos φ] sin (kx − ωt +
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1
φ)
2
interference
Interference of Waves
If two sinusoidal waves of same amplitude and
wavelength travel in the same direction, they interfere
to produce a resultant sinusoidal wave
fully constructive
interference
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fully destructive
interference
Phasors
We can represent a string wave vectorially with a phasor:
magnitude of the phasor = the amplitude of the wave
rotates around the origin
angular speed of the phasor = angular frequency ω of the wave
The wave
y1 (x, t) = ym1 sin (kx − ωt)
is represented by the phasor shown in left figure (top)
And the wave
y2 (x, t) = ym2 sin (kx − ωt + φ)
is represented by the phasor shown in left figure (middle)
The combined wave is
!
y ! (x, t) = ym
sin (kx − ωt + β)
and you add two waves vectorially in the phasor diagram
Fundamentals of Physics by Eunil Won, Korea University
Standing Waves
If two sinusoidal waves of the same amplitude and wavelength travel in opposite directions
along a stretched string, their interference with each other produces a standing wave
There are places along the string, called nodes where the string never moves
antinodes: halfway between nodes, the amplitude of the resultant wave is a maximum
Fundamentals of Physics by Eunil Won, Korea University
Standing Waves
We represent two combining waves :
The principle of
superposition gives:
y1 (x, t) = ym sin (kx − ωt)
y2 (x, t) = ym sin (kx + ωt)
y ! (x, t) = y1 (x, t) + y2 (x, t) = ym sin (kx − ωt) + ym sin (kx + ωt)
y ! (x, t) = [2ym sin kx] cos ωt
1
1
sin α + sin β = 2 sin (α + β) cos (α − β)
2
2
The standing wave has zero amplitude when sin kx = 0 :
kx = nπ,
for n = 0, 1, 2, ...
substituting k = 2π /λ
gives
λ
x=n ,
2
for n = 0, 1, 2, ...
(nodes)
The standing wave has maximum amplitude when |sin kx | = 1 :
1
kx = (n + )π,
2
for n = 0, 1, 2, ...
Fundamentals of Physics by Eunil Won, Korea University
or x =
!
1
n+
2
"
λ
,
2
for n = 0, 1, 2, ...
(antinodes)
Standing Waves
Reflections at Boundary:
Fixed end: the reflected and incident waves have
opposite signs making standing wave
Free to slide: the reflected and incident waves have
same signs
fixed end
free to slide
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Standing Waves and Resonance
For certain frequencies, the
interference produces a standing
wave pattern (for strings with
two ends clamped) : resonance
When
λ=2L
λ=L
λ = 2/3 L...
resonance occurs
(L : length of the string)
Thus, a standing wave can be set up on a
string of length L
2L
λ=
n
for n = 1, 2, 3, ...
resonance frequency:
v
v
f = =n
λ
2L
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for n = 1, 2, 3, ...
Summary
Transverse wave: the displacement of the oscillating element is perpendicular
to the direction of travel of the wave
Longitudinal wave: the displacement of the oscillating element is parallel to the
direction of travel of the wave
sinusoidal waves:
angular frequency:
wave speed:
Fundamentals of Physics by Eunil Won, Korea University
y(x, t) = ym sin (kx − ωt)
ω
1
=
f=
T
2π
2π
ω=
T
v=
frequency:
!
τ
µ