16
W
Waves
I
Waves are of 3 main types:
1. Mechanical waves
W t
Water,
sound,
d seismic
i i waves (地震波) - governed
d
by Newton’s laws, and exist only within a material
medium.
2. Electromagnetic waves.
Visible and UV light, radio waves, microwaves, x
rays, radar waves – require no material medium to
exist, through the vacuum of space to reach us. All
EM waves travel through a vacuum at c ~3×10
3×108 m/s.
m/s
„
1
3 Matter waves.
3.
waves
Although these waves are commonly used in
modern technology,
gy, theyy are p
probablyy veryy
unfamiliar to you. These waves are associated with
electrons, protons, and other fundamental
particles and even atoms and molecules.
particles,
molecules Because
we commonly think of these particles as
constituting matter, such waves are called matter
waves.
2
transverse wave (物質元素の
位移⊥波傳播方向）
„
„
longitudinal wave (物質元素
の位移//波傳播方向）
Both a transverse wave and a longitudinal wave
are said to be traveling waves (行進波) because
th both
they
b th travel
t
l from
f
one point
i t to
t another.
th
Note that it is the wave that moves from end to
end not the material (string or air) through which
end,
the wave moves.
3
„
At time t, the transverse displacement y of the
material element (物質元素) located at position x
is given by
y(x,t)
x
4
5
„
The wavelength λ of a wave is the distance
b t
between
repetitions
titi
off th
the shape
h
off the
th wave (or
(
wave shape).
Fixing time (let t=0)
6
„
The displacement y is the same at x=x1 and x=x1+λ
„
A sine function begins to repeat itself when its angle
is increased by 2π rad,
rad so in Eq.
Eq 16-4
16 4 we must have
kλ=2π, or
We call k the angular wave number of the wave.
7
Fixing position (let xx=0)
0)
8
This can be true only if ωT=2π, or if
We call the angular frequency of the wave.
„
The frequency f of a wave is defined as 1/T and is
related to the angular frequency ω by
„
We can generalize
W
li
E 16-2
Eq.
16 2 by
b inserting
i
ti
a phase
h
constant (初相角:x=0且t=0時の相位角) in wave
function:
9
„
„
The wave is traveling in +x direction, the entire wave
pattern moving a distance Δx in that direction during
the interval Δt.
Δt The ratio Δx/Δt (or dx/dt) is the
wave speed v. How can we find its value?
If p
point A retains ((保持)) its displacement
p
as it moves,,
the phase in Eq. 16-2 giving it that displacement
must remain a constant, that is,
10
To find the wave speed v, the derivative of Eq. 16-11
„
Using Eq. 16-5 (k=2π/λ) and Eq. 16-8 (ω=2π/T),
we can rewrite the wave speed as
The equation v=λ/T tells us that the wave speed
is one wavelength per period.
period
11
„
Eq. 16-2 describes a wave moving in +x direction.
We can find the equation of a wave traveling in the
opposite direction by replacing t in Eq. 16-2 with -t.
This corresponds to the condition
„
Thus, a wave traveling in the -x direction is
described by the equation
You will find for its velocity
12
„
Consider now a wave of arbitrary shape, given by
where h represents any function of kx±ωt,
kx±ωt the
sine function being one possibility.
„
All traveling waves must be of the form of Eq. 16-17.
/
Thus y(x,t)=(ax+bt)
Thus,
y(x t)=(ax+bt)1/2
represents a possible
traveling wave. The function y(x,t)=sin(ax2-bt) does
not represent
p
a traveling
g wave.
Secs.16-6 and 16-7: self-study
13
z
z
As a wave passes through any element on a
stretched string, the element (dm) moves
perpendicularly to the wave
wave’ss direction of travel.
travel
By applying Newton’s 2nd law to the element’s
motion,, we can derive a g
general differential
equation, called the wave equation, that governs
the travel of waves of any type.
dm
14
„
Newton’s 2nd law written for y-components
(Fy=ma
may) gives us
„
Consider the element can have only a slight tilt,
ℓ~dx
15
where v is the wave speed on a stretched string
繩波波速
μ: linear density of the string; τ: tension in
the string
16
z
z
Suppose that two waves travel simultaneously
along the same stretched string. Let y1(x,t) and
y2(x,t) be the displacements that the string would
experience if each wave traveled alone. The
displacement of the string when the waves overlap
is then the algebraic sum
This summation of displacements along the string
means that
th t
Overlapping waves algebraically add to produce a
resultant wave (or net wave).
17
„
„
The
principle
of
superposition,
iti
which
hi h says
that when several effects
occur ssimultaneously,
occu
u ta eous y, ttheir
e
net effect is the sum of the
individual effects.
Wh
When
th pulses
the
l
overlap,
l
th
the
resultant pulse is their sum.
Moreover, each pulse moves
through the other as if the
other were not present:
Overlapping waves do not in
any way alter the travel of
each other.
18
Suppose we send two sinusoidal waves of the
same wavelength and amplitude in the same
direction
along
a
stretched
string.
The
superposition principle applies. What resultant
wave does it predict for the string?
z The resultant wave depends on the extent (程度)
to which the waves are in phase ( 同 相 ) with
respect to each other — that is, how much one
waveform is shifted from the other waveform.
(1) If the waves are exactly in phase,
phase they combine to
double the displacement.
((2)) If tthey
ey a
are
ee
exactly
act y out o
of p
phase,
ase, tthey
ey co
combine
b e to
cancel everywhere, and the string remains straight.
z
19
„
„
We call this phenomenon of combining waves
interference (干涉).
(干涉)
Let one wave traveling along a stretched string be
given
g
e by
and another, shifted from the first, by
They differ only by a constant angle φ, the phase
constant of wave 2.
2 These waves are said to be out
of phase by φ or to have a “phase difference (相差)
of φ
φ”,, or one wave is said to be phase
phase-shifted
shifted (相位
移) from the other by φ.
20
„
From the principle of superposition (Eq. 16-46),
the resultant wave is the algebraic sum of the two
interfering waves and has resultant displacement
By
We obtain
21
22
„
The resultant wave differs from the two interfering
g
waves in two respects: (1) its phase constant is φ/2,
and (2) its amplitude ym’ is
If φ=0o, the two interfering waves are exactly in
phase, as in Fig. 16-16a. Then Eq. 16-51 reduces to
This resultant wave is plotted in Fig.
Fig 16
16-16d
16d.
Interference that produces the greatest possible
amplitude
p
is
called
fully
y
constructive
interference (完全建設性干涉).
23
„
If φ=180o, the interfering waves are exactly out of
phase (完全反相) as in Fig.
Fig 16-16b.
16 16b Then cosφ/2
becomes 0, and the amplitude of the resultant
wave (Eq. 16-52) is zero. We then have
The resultant wave is plotted in Fig. 16-16e. This
type of interference is called fully destructive
i
interference
f
(完全破壞性干涉)
(完全破壞性干涉).
Note that when interference is neither fully
constructive nor fully destructive,
destructive it is called
intermediate interference ( 中 等 程 度 の ). The
amplitude
p
of the resultant wave is then
intermediate between 0 and 2ym [Figs. 16.16c, f].
24
25
(Assume ym1=ym2=ym)
26
z
z
We can represent any other type of wave vectorially
with a phasor (相量).
A phasor is a vector
t that has a magnitude equal to
the amplitude of the wave and that rotates around
an origin.
g
The angular
g
speed
p
of the p
phasor = The
angular frequency ω of the wave.
Because waves y1 and y2 have the same angular
wave number k and angular frequency ω, we know
from Eqs.
q 16-51 that their resultant is of the form
27
-ωt
ωt
-ωt+φ
φ
y'm = y m1 + y m 2 − 2 y m1 y m 2 cos(π − φ)
2
tan β =
-ωt+β
ωt+β
2
2
y m 2 sin φ
y m1 + y m 2 cos φ
若已知 ym1, ym2與φ，可求出合
成位移y’之振幅y’m與相角β
28
z
z
z
In Sec. 16-10, we discussed two sinusoidal waves
g and amplitude
p
traveling
g in
of the same wavelength
the same direction along a stretched string.
What if they travel in opposite directions? We can
again
i find
fi d the
th resultant
lt t wave by
b applying
l i
th
the
superposition principle.
The outstanding feature of the resultant wave is
that (1) there are places along the string, called
nodes (節點), where the string never moves.
(2) Halfway between adjacent nodes are
antinodes (反節點), where the amplitude of the
resultant wave is a maximum.
maximum
29
„
Wave patterns such as that of Fig. 16-19c are called
standing waves (駐波) because the wave patterns
d not move left
do
l f or right
h ( 駐 留 在 某 區 域 );
) the
h
locations of the maxima and minima do not change.
入射波
反射波
Fig. 16-19
30
If two sinusoidal waves of the same amplitude and
wavelength
g
travel in opposite
pp
directions along
g a
stretched string, their interference with each other
produces a standing wave.
To analyze a standing wave, we represent the two
combining waves with the equations
The principle of superposition gives
31
Applying the trigonometric relation of Eq. 16
16-50
50 leads
to
This equation is different to the traveling wave
function [Eq. (16.17)].
32
„
„
In a traveling sinusoidal wave, the amplitude of the
wave is the same for all string elements.
elements That is not
true for a standing wave, in which the amplitude
varies with position x.
In the standing wave of Eq. 16-60, the amplitude is
0 for values of sinkx=0. Those values are
Substituting
g k=2π/λ
/ in this equation,
q
, we g
get
which are the positions of zero amplitude—the nodes.
Note that adjacent nodes are separated by λ/2.
33
„
The amplitude of the standing wave of Eq. 16-60
h
has
a maximum
i
value
l
off 2y
2 m which
hi h occurs for
f
values of |sinkx|=1. Those values are
Substituting k=2π/λ in Eq. 16-63, we get
as the positions of maximum amplitude — the antinodes.
34
„
„
[Fig. 16-21a]
[Fig
16 21a] In a “hard”
reflection, there must be a node
at the support
pp
because the string
g
is fixed there. The reflected and
incident
pulses
must
have
opposite signs,
signs so as to cancel
each other at that point.
[[Fig.
g 16-21b]] In a “soft” reflection,,
the incident and reflected pulses
reinforce ( 增 強 ) each other,
creating an antinode at the end
of the string (not fixed there); the
maximum displacement of the
ring
i
i twice
is
i
the
h amplitude
li d off
either of these pulses.
Hard (fixed)
(Soft) Not fixed
Node
Antinode
35
z
[Fig. 16-22] Consider a string, such as a guitar string,
that is stretched between two clamps.
clamps Suppose we
send a continuous sinusoidal wave of a certain
frequency along the string, say, toward the right.
Wh the
When
th wave reaches
h the
th right
i ht end,
d it reflects
fl t and
d
begins to travel back to the left. That left-going wave
then overlaps
p the wave that is still traveling
g to the
right. When the left-going wave reaches the left end,
it reflects again and the newly reflected wave begins
to travel to the right,
right overlapping the left-going
left going and
right-going waves. In short, we very soon have many
overlapping traveling waves, which interfere with
one another.
36
„
„
[Fig. 16-22] For certain frequencies, the interference
produces a standing
p
g wave p
pattern ((or oscillation
mode). Such a standing wave is said to be produced
at resonance, and the string is said to resonate at
these certain frequencies,
frequencies
called resonant
frequencies.
If the stringg is oscillated at some frequency
q
y other than
a resonant frequency, a standing wave is not set up.
Then the interference of the right-going and leftgoing traveling waves results in only small (perhaps
imperceptible察覺不出的) resultant oscillations of the
string.
37
Fig. 16-22 Stroboscopic (頻閃觀測儀的) photographs reveal (imperfect)
Fig
standing wave patterns on a string being made to oscillate by an
oscillator at the left end. The patterns occur at certain frequencies of
oscillation.
oscillation
P. 37
38
„
„
[Fig.
16-23]
To
find
expressions for the resonant
frequencies of the string, we
note that a node must exist
att each
h off its
it ends,
d because
b
each end is fixed and cannot
oscillate.
A standing wave can be set
up on a string of length L by
a wave with
ith a wavelength
l
th
equal to one of the values
n=1
n=2
n=3
Fig. 16-23
39
The resonant frequencies that correspond to these
wavelengths
g
follow from Eq.
q 16-13:
„
„
Eq. 16-66 tells us that the resonant frequencies are
integer multiples of the lowest resonant frequency,
frequency
flowest=v/2L, which corresponds to n=1. The oscillation
mode with that lowest frequency is called the
fundamental mode (基模) or first harmonic (第一諧波).
The second harmonic is the oscillation mode with n=2,
the third harmonic is that with n=3,
n=3 and so on.
on The
frequencies associated with these modes are often
possible
labeled f1, f2, f3, and so on. The collection of all p
oscillation modes is called the harmonic series, and n
is called the harmonic number of the nth harmonic. 40
The p
powder collects the
nodes, which are circles
and straight lines in this
2D example.
41
42
43
44
察覺不出的
45
46
47
48
Homework
Ans:
49
Ans:
50
Ans:
51
Ans:
52
Ans:
53
Extra
Ans: 0.25m
54
Ans: (a) ‐0.039 m (b) 0.15 m (c) 0.79 m‐1 (d) 13 rad/s (e) a plus sign in front
of ω (f) ‐0.14 m
55