Transcript 幻灯片 1 - Yangtze University

Fundamentals of Physics
Mechanics
(Bilingual Teaching)
张昆实
School of Physical Science and Technology
Yangtze University
Chapter 17 Waves
17-1 Waves and Particles
17-2 Types of Waves
17-3 Transverse and Longitudinatl Waves
17-4 Wavelength and Frequency
17-5 The Speed of a Traveling wave
17-6 Wave Speed on o Stretched String
17-7 Energy and Power of a Traveling
String
Chapter 17 Waves
17-8 The Principle of Superposition for
Waves
17-9 Interference of Waves
17-10 Phasors
17-11 Standing Waves
17-12 Standing Waves and Resonance
17-1 Waves and Particles
Every day we use waves in one
form or another. Waves carry
energy.
An alarm clock wakes you up
because it emits sound waves
when it ‘ goes off ’.
The television set picks up
signals in the form of radio
waves and transforms them
into a picture, which is
carried to your eye by light
waves. In these examples
waves are being used to
send information.
17-1 Waves and Particles
In a wave, information and energy move
from one point to another but no material
object makes that journey.
Particle and wave are the two great
concepts in classical physics, in the sense
that we seem able to associate almost every
branch of the subject with one or the other.
17-1 Waves and Particles
Particle and wave are the two great concepts in
classical physics. The two concepts are quite
different.
The word particle suggests a tiny concentration of
matter capable of transmitting energy.
The word wave suggests just the opposite
namely, a broad distribution of energy, filling the
space through which it passes.
The job at hand is to put aside particles for a
while and to learn something about waves.
17-2 Types of Waves
Waves are of three main types:
1. Mechanical waves.
These waves are most familiar because we
encounter them almost constantly;
common examples include water
waves, sound waves, and seismic
waves. All these waves have
certain central features: They are
governed by Newton's laws, and
they can exist only within a
material medium, such as water,
air, and rock.
17-2 Types of Waves
2. Electromagnetic waves.
These waves are less familiar, but you use them
constantly;
common examples include visible and ultraviolet light,
radio and television waves, microwaves, x rays, and radar
waves. These waves require no material medium to exist.
All electromagnetic waves travel through a vacuum at the
same speed c = 299 792 458 m/s (17-1)
17-2 Types of Waves
3. Matter waves.
Although these waves are commonly used in modern
technology, their type is probably very unfamiliar to you.
These waves are associated with electrons,
protons, and other fundamental particles, and
even atoms and molecules. Because we
commonly think of these things as constituting
matter, such waves are called matter waves.
Much of what we discuss in this chapter applies
to waves of all kinds. However, for specific
examples we shalI refer to mechanical waves.
17-3 Transverse and Longitudinatl Waves
A wave sent along a stretched, taut string is the
simplest mechanical wave.
If you move your hand up and
down in continuous simpIe
harmonic motion, a continuous
wave travels along the string at
velocity V.
The displacement of every oscillating string
element is perpendicular to the direction of travel
of the wave. as indicated in Fig. 17-1. This motion
is said to be transverse, and the wave is said to
be a transverse wave.
17-3 Transverse and Longitudinatl Waves
Figure 17-2 shows how a sound wave can be produced by
a piston in a long, air-filled pipe.
If you suddenly move the piston
rightward and then leftward, you
can send a pulse of sound along
the pipe.
If you push and pull on the piston in
simple harmonic motion,
a sinusoidal wave travels along the pipe. Because
the motion of the elements of air is parallel to the
direction of the wave's travel, the motion is said to
be longitudinal, and the wave is said to be a
longitudinal wave.
17-4 Wavelength and Frequency
To completely describe a wave on a
string, we need a function that gives
the shape of the wave.
Imagine a sinusoidal wave sweeps
through succeeding elements of the
string, the elements oscillate parallel to the y axis. At time t,
the displacement y of the element located at x is given by
y( x, t )  ym sin(kx  t )
(17-2)
Because this equation is written in terms of position x, it
can be used to find the displacement of all the elements of
the string as a function of time. Thus, it can tell us the
shape of the wave at any given time and how that shape
changes as the wave moves along the string.
17-4 Wavelength and Frequency
y( x, t )  ym sin(kx  t )
(17-2)
A sinusoidal wave is traveling along
the +x axis. A red arrow pointing to a
high point of the wave moves to the
right with the wave shape.
valley
But the string moves only
parallel to the y axis. Look at
the red elements at x=0 in the
five snapshots, its displacement y  0,  ym , 0, ym , 0
in each of the snapshots,
respectively.
peak
17-4 Wavelength and Frequency
y( x, t )  ym sin(kx  t )
(17-2)
Amplitude and Phase
The amplitude ym of a wave is the magnitude of the maximum displacement
of the elements from their equilibrium
positions as the wave passes through
them. ym is always a positive quantity.
The phase of the wave is the argument kx  t of the sine in Eq.17-2.
As the wave sweeps through a string
element at a particular position x, the
phase changes linearly with time t.
valley
peak
17-4 Wavelength and Frequency
Wavelength and Angular Wave
Number
The wavelength
of a wave is the
distance between the repetition of the
shape of the wave. (marked in Fig. a)
at t=0, Eq. y( x, t )  ym sin(kx  t ) gives

y ( x, 0)  ym sin kx
peak
valley
(17-3)
A sine function begins to
By definition, the displacement y repeat itself when its angle
is the same at both ends of this is increased by 2 rad :
wavelength, that is at x  x1 and ym sin kx1  ym sin(kx1  2 )
x  x1   Thus by Eq.17-3
We have
have k   2 ,
We
ym sin kx1  ym sin k ( x1   )
 ym sin(kx1  k  ) (17-4)
or k 
2

SI unit:
k   2 ,
(angular wave (17-5)
number)
rad / m
17-4 Wavelength and Frequency
Period, Angular frequency
and frequency
The period of oscillation T of a
wave to be the time any string
element takes to move through
one full oscillation.
t1  T
t1
Take a element at x=0, you would results yield
see that the single element of the  ym sin t1   ym sin  (t1  T )
string at that position moves up
(17-7)
  ym sin(t1  T )
and down in SHM given by
This can be true only if
y(0, t )  ym sin(t )
T  2 or if
  ym sin t
(17-6)
Applying Eq.17-6 to both ends
of period T and equating the
2

T
SI unit:
(angular
frequency)
rad / s
(17-8)
17-4 Wavelength and Frequency
Period, Angular frequency
and frequency
t1
t1  T
The frequency f of a wave is
defined as 1/T and is related to
the angular frequency  by
1 
f  
T 2
(frequency)
(17-8)
Like the frequency of SHM, this frequency f is a
number of oscillation per unit time ---- here, the
number made by a sigle element as the wave
moves through it.
Frequency f is measured in hertz or kilohertz
17-5 The Speed of a Traveling wave
Figure17-7shows two snapshots
of wave of Eq.17-2. The entire
wave pattern moving a distance
of x in the positive direction of
x during the interval t .
y( x, t )  ym sin(kx  t )
(17-2)
The wave speed is
x
dx
v  lim

t  0 t
dt
wave form do.) If point A
retains its displacement
Find the value of the wave speed as it moves, the phase in
As the wave moves, each point
Eq.17-2 giving it that disof the moving wave form retains placement must remain a
its displacement y. (Points on the constant:
string do not retain their
kx  t  a constant (17-10)
displacement, but points on the
17-5 The Speed of a Traveling wave
kx  t  a constant
y( x, t )  ym sin(kx  t )
(17-10)
(17-2)
Although this argument is constant, both x and t are changing.
In fact, as t increases, x must
also, to keep the argument
constant. This confirms that the
wave pattern is moving in the
2
positive direction of x.
2
Using k 
and  
To find the wave speed v, take
T

the derivative of Eq.17-10
 
or
dx
k
  0
dt
dx

v
dt
k
v
(17-11)
k

T
f
(17-12)
( wave speed )
The wave speed is one
wavelength per period.
17-5 The Speed of a Traveling wave
If a wave moves to the negative
direction of x, using –t to replace
t in Eq.17-2, to get the wave
equation for that case :
y( x, t )  ym sin(kx  t )
y( x, t )  ym sin(kx  t )
(17-2)
(17-14)
Similarly the argument
kx  t  a constant
(17-13)
The wave speed in the
opposite direction of x is then
dx

v
dt
k
(17-15)
The minus sign verifies that
the wave is indeed moving
in the negative direction of
x and justifies our
switching the sign of the
time variable.
17-6 Wave Speed on o Stretched String
The speed of a wave is related to the wave's wavelength
and frequency by Eq. 17-12 v   f , but it is set by the
properties of the medium.
the medium's mass and elasticity properties determine
how fast the wave can travel in the medium. Conversely,
it should be possible to calculate the speed of the wave
through the medium in terms of these properties.
Dimensional Analysis
In dimensional analysis we carefully examine the
dimensions of all the physical quantities that enter into a
given situation to determine the quantifies they produce.
In this case, we examine mass and elasticity to find a
speed v, which has the dimension of length divided by
time, or LT-1.
17-6 Wave Speed on o Stretched String
Dimensional Analysis
The dimension of the wave speed v :
LT -1
For the mass, we use the mass of a string element, which
is represented by the mass m of the string divided by the
length l of the string. We call this ratio the linear density 
of the string. Thus,   m / l , its dimension being mass
divided by length, ML-1.
You cannot send a wave along a string unless the string is
under tension, which means that it has been stretched and
pulled taut by forces at its two ends. we can associate the
tension in the string with the stretching (elasticity) of the
string. The tension and the stretching forces it produces
have the dimension of a force — namely, MLT-2 (from F =
ma).
17-6 Wave Speed on o Stretched String
Dimensional Analysis
the dimension of the wave speed
v
:
LT -1
the dimension of the linear density 
:
ML-1

:
MLT-2
the dimension of the tension and
the stretching forces
The goal here is to combine  and  in such a way as
to generate v . A little juggling of various combinations
suggests

vC

(17-21)
in which C is a dimensionless constant that cannot be
determined with dimensional analysis. In our second
approach to determining wave speed, you will see that
Eq. I7-21 is indeed correct and that C = 1.
17-6 Wave Speed on o Stretched String
Derivation from Newton's Second Law
Consider a single symmetrical
pulse such as that of Fig. 17-9,
moving from left to right along a
string with speed v. For
convenience, we choose a reference frame in which the pulse
remains stationary; that is, we run along with the pulse,
keeping it constantly in view. In this frame, the string
appears to move past us, from right to left in Fig. 17-9. with
speed v.
Consider a small string element within the pulse, of length
l l forming an arc of a circle of radius R and subtending an

angle 2 at the center of that circle.
17-6 Wave Speed on o Stretched String
Derivation from Newton's Second Law
A force with a magnitude equal to
the tension in the string pulls
tangentially on this element at
each end. The horizontal components of these forces cancel, but
the vertical compenents add to
form a radial restoring force . In magnitude,
l
F  2( sin  )   (2 )  
R
The mass of the element m  l
(force)
(17-22)
(mass)
(17-23)
At the moment shown in Fig~ 17-9, the swing element Al is
moving in an arc of a circle. Thus, it has a cengilsital
acceleration toward the center of that circle, given by
17-6 Wave Speed on o Stretched String
Derivation from Newton's Second Law
l
F 
R
m  l
(force)
(17-22)
(mass)
(17-23)
The centripital acceleration of
the arc
v2
a
(acceleration) (17-24)
R
From Newton's second law
l
v

 ( l )
R
R
2
F  ma
Solving this equation for the speed v yields
v


(speed)
(17-25)
17-6 Wave Speed on o Stretched String
Derivation from Newton's Second Law
l
F 
R
m  l
(force)
(17-22)
(mass)
(17-23)
The centripital acceleration of
the arc
v2
a
(acceleration) (17-24)
R
The speed of a wave
From Newton's second law F  ma
along a stretched
2
l
v
ideal string depends

 ( l )
only on the tension
R
R
Solving this equation for the speed v and linear density of
the string and not on
yields

the frequency of the
v
(17-25)
(speed)

wave.
17-7 Energy and Power of a Traveling String wave
When we set up a wave on a stretched
string, we provide energy for the
motion of the string. As the wave
moves away from us, it transports that
energy as both kinetic energy and
elastic potential energy.
Kinetic Energy
An element of the string of mass dm , oscillating transversely in SHM as the wave passes through it, has kinetic
energy associated with its transverse velocity u . When the
element is rushing fiuongh its y = 0 position (element b in
Fig. 17-11), its transverse velocity — and thus its kinetic
enexgy — is a maxtmum. When the element is at its extreme
position y =ym (as is element a), its transverse velocity —
and thus its kinetic energy — is zero.
17-7 Energy and Power of a Traveling String wave
Elastic Potential Energy
To send a sinusoldal wave along a previously straight string, the wave must
necessarily stretch the siring. As a
string element of length dx oscillates
transversely, its length must increase
amd decrease in a periodic way if the
dm
string element is to fit the sinusoidol
wave form. Elastic potential energy is associated with these
length changes, just as for a spring.
When the string element is at its y = ym position, its length
has its normal undisturbed value dx, so its elastic potential
energy is zero. However, when the element is rushing
through its y = 0 position, it is stretched to its maximum
extent, and its elastic poteatial energy then is a maximum.
17-7 Energy and Power of a Traveling String wave
Energy Transport
The oscillating string element at
y = 0 : has maximum Ekin  U ela
y=ym : has no energy
u
dm
As the wave travels along the x axis,
forces due to the tension in the string
continuously do work to transer energy
from regions with energy to regions with no energy.
As the string sections oscillate perpendicutarly to the x axis,
they have kinetic energy and elastic potential energy. As the
wave moves into sections that were previously at rest,
energy is transferred into those new sections. Thus, we say
that the wave transports the energy along the string.
17-7 Energy and Power of a Traveling String wave
The Rate of Energy Transmission
The kinetic energy of the string element
dk  12 dm  u 2
Find the transverse speed u, differentiate y( x, t )  ym sin(kx  t ) (17-2)
With respect to time while holding
constant:
y
u
u
dm
x
  ym cos(kx  t )
t
so dK  1 (  dx)(  ym ) 2 cos 2 ( kx  t )
2
dK 1
 2  v 2 ym2 cos 2 (kx  t )
dt
u
(17-27)
(17-28)
(17-29)
The average rate at which kinetic energy is transported is
 dK 
2 2
2
2 2
1
1




v

y
cos
(
kx


t
)


v

ym
m 


2
4

avg
 dt avg
(17-30)
17-7 Energy and Power of a Traveling String wave
The Rate of Energy Transmission
The average rate at which kinetic
energy is transported is
 dK 
2 2
1


v

ym


4
 dt avg
(17-30)
Elastic potential energy is also carried
along with the wave, and at the same
average rate given by Eq. 17-30.
u
dm
u
 dU 
2 2
1
Although we shall not examine


v

ym


4
the proof, you should recall that,
 dt avg
power : the average
in an oscillating system such as average
rate at which energy of both kinds
is transmitted by the wave is
a pendulum or a spring block
system, the average kinetic
 dK 
Pavg  2    12  v 2 ym2
energy and the average potential
 dt avg (17-32)
energy are indeed equal.
(17-31)
17-8 The Principle of Superposition for Waves
It often happens that two or more
waves pass simultaneously through
the same region.
When we listen to a concert, for example,
sound waves from many instruments
fall simultaneously on our eardrums.
The electrons in the antennas of our
radio and television receivers are set in
motion by the net effect of many
electromagnetic waves from many
different broadcasting centers.
The water of a lake or harbor may be
churned up by waves in the wakes of
many boats.
wave1
wave2
wave1
wave2
wave1
u
wave2
wave2
wave2
wave1
wave1
17-8 The Principle of Superposition for Waves
Suppose that two waves travel simultaneousIy along the same stretched
string. Let y1(x, t) and y2(x, t) be the displacements that the string would
experience if each wave traveIed alone.
The displacement of the string when the
waves overlap is then the algebraic sum
y( x, t )  y1 ( x, t )  y2 ( x, t )
wave1
wave2
(17-33)
This summation of displacements
along the string means that
Overlapping waves algebraically add to
produce a resultant wave (or net wave).
wave2
wave1
This is another example of the principle of superposition,
which says that when several effects occur simultaneously,
their net effect is the sum of the individual effects.
17-8 The Principle of Superposition for Waves
the principle of superposition
y( x, t )  y1 ( x, t )  y2 ( x, t )
wave1
wave2
(17-33)
Figure 17-12 shows a sequence of
snapshots of two pulses traveling in
opposite directions on the same
stretched string. When the pulses
overlap, the resultant pulse is their
sum. Moreover, each pulse moves
through the other, as if the other were
not present.
wave1
wave2
Overlapping waves do not in any way alter the travel of
each other.
17-9 Interference of Waves
Suppose we send two sinusoidal waves of the same
wavelength and amplitude in the same direction along a
stretched string. The superposition principle applies.
The resultant wave depends on the extent to which the
waves are in phase (in step) with respect to each other—
that is, how much one wave form is shifted from the other
wave form.
If the waves are exactly in phase (so that the peaks and
valleys of one are exactly aligned with those of the other),
they combine to double the displacement of either wave
acting alone.
The interference of Waves
If they are exactly out of phase (the peaks of one are exactly
aligned with the valleys of the other), they combine to
cancel everywhere, and the string remains straight.
17-9 Interference of Waves
Let two wave traveling along a stretched string be given by
y1 ( x, t )  ym sin(kx  t )
y2 ( x, t )  ym sin(kx  t   )
(17-34)
(17-35)
These waves have the same angular frequency  (and thus
the same frequency f), the same angular wave number k
(and thus the same wavelength ), and the same amplitude
ym .They both travel in the positive direction of the x axis,
ii
with the same speed, given by v    . They differ only by
a constant angle  , which we call the phase constant.
From the principle of superpeaifion, the resultant wave is
y( x, t )  y1 ( x, t )  y2 ( x, t )
 ym sin(kx  t )  ym sin(kx  t   )
(17-36)
17-9 Interference of Waves
From the principle of superpeaifion, the resultant wave is
y( x, t )  y1 ( x, t )  y2 ( x, t )
 ym sin(kx  t )  ym sin(kx  t   )
(17-36)
Applying trigonometric equation
sin   sin   2sin 12 (   ) cos 12 (   )
(17-37)
y( x, t )  [2 ym cos 12  ]sin(kx  t  12  )
(17-38)
Amplitude
Oscillating term
The resultant wave is also a sinusoidal wave traveling in
the dliection of increasing x.
If two sinusoidal waves of the same amplitude and wavelength travel in the same direction along a stretched string,
they interfere to produce a resultant sinusoidal wave
traveling in that direction.
17-9 Interference of Waves
y( x, t )  [2 ym cos 12  ]sin(kx  t  12  )
Amplitude
ym  2 ym cos 12 
If   0
Oscillating term
(17-38)
Phase constant
(17-39)
full constructive

y ( x, t )  2 ym sin(kx  t ) interference
(17-40)
The two intefering waves are exactly in phase. the
amplitude of the resultant wave is twice the amplitude of
either interfering wave. That is the greatest amplitude the
resultant wave can have, because cos0  1
If
   rad
or 1800
y( x, t )  0
full destructive
interference
(17-41)
The interfering waves are exactly out of phase as in
Fig. 17-14b. Then cos 12  becomes cos 12   0,and the
amplitude of the resultant wave is zero.
17-9 Interference of Waves
 0
exactly in phase
full constructive
interference
   rad
  23  rad
exactly out of phase
full destructive
interference
intermediate
interference
17-9 Interference of Waves
Because a sinusdidal wave repeats its shape every 2 rad,
a phase difference   2 rad (or 3600) corresponds to a
shift of one wave relative to the other wave by a distance
equivalent to one wavelength. Thus, phase differences can
be described in terms of wavelengths as well as angles.
17-10 Phasors
phasor
Phasor 2
Phasor 1
Phasor 2
y’
Phasor 1
We can represent a string wave (or any
other type of wave) vectorially with a
phasor. In essence, a phasor is a vector
that has a magnitude equal to the
amplitude of the wave and that retates
around an origin; the angular speed of the
phasor is equal to the angular frequency
 of the wave. For example, the waves
y1 ( x, t )  ym1 sin(kx  t )
(17-42)
y2 ( x, t )  ym 2 sin(kx  t   )
(17-43)
are represented by phasor 1 and phasor 2.
because the phasors rotate at the same
angular speed  , the angle between the
two phasors is always  ,
17-10 Phasors
y1 ( x, t )  ym1 sin(kx  t )
y2 ( x, t )  ym 2 sin(kx  t   )
phasor
(17-42)
(17-43)
Because waves y1 and y 2 have the same
angular wave number k and angular frequency  ,their resultant is of the form
y( x, t )  ym sin(kx  t   )
Phasor 2
Phasor 1
Phasor 2
y’
Phasor 1
(17-44)
 is the is the amplitude of the rewhere ym
sultant wave and  is its phase constant.
To find the values we vectofially add the
two phasors at any instant during their
rotation, as in Fig. 17-15c where phasor
ym 2 has been shifted to the head of
phasor ym1. The magnitude of the vector
sum equals the amplitude. the angle
between the vector sum and the phasor
for y1 equals the phase constant  .
17-11 Standing Waves
Demonstration of the standing waves
17-11 Standing Waves
node
antinode
Two sinusoiclal waves of the
same wavelength and amplitude are traveling in the opposite direction along a stretched
string.
y1 ( x, t )  ym sin(kx  t )
(17-45)
y2 ( x, t )  ym sin(kx  t ) (17-46)
Feature of the standing waves The principle of superposition
There are nodes, where the
gives, for the combined wave
string never moves.
y( x, t )  y1 ( x, t )  y2 ( x, t )
there are antinodes between  ym sin(kx  t )  ym sin(kx  t )
adjacent nodes, where the
Applying trigonometric relation:
string has its maximum
y( x, t )  [2 ym sin kx]cos t
amplitude.
17-11 Standing Waves
node
amplitude
maximum
zero
| sin kx | 1
sin kx  0
kx  n
antinode
n  0,1, 2,...
(17-48)
Feature of the standing waves
y( x, t )  [2 ym sin kx]cos t
Take the absolute value of
the quantity 2 ym sin kx in the
brackets to be the amplitude
at x . The amplitude varies
with position.
k  2 
xn

2
n  0,1, 2,...
kx  (n  12 )
n  0,1, 2,...
(17-50)
k  2 
1

x  n 
2 2

n  0,1, 2,...
(17-49)
(17-51)
(nodes)
(antinodes)
17-11 Standing Waves
Incident
wave
Reflections at a boundary
We can set up a standing wave in a
stretched string by allowing a traveling wave to be reflected from the far
end of the string so that it travels
back through itself.
The incident (original) wave and the
reflected wave can then be described
by Eqs. 17-45 and 17-46, respectively,
and they can combine to form a
pattern of standing waves.
reflected
wave
In Fig. 17-19, we use a single pulse
to show how such reflections take
place.
17-11 Standing Waves
Incident
wave
reflected
wave
In Fig. 17-19a, the string is fixed at its
left end. When the pulse arrives at
that end, it exerts an upward force on
the support (the wall). By Newton's
third law, the support exerts an opposite force of equal magnitude on the
string. This second force generates a
pulse at the support, which travels
back along the string in the direction
opposite that of the incident pulse, In
a "hard" reflection of this kind, there
must be a node at the support
because the string is fixed there. The
reflected and incident pulses must
have opposite signs, so as to cancel
each other at that point.
17-11 Standing Waves
Incident
wave
Incident
wave
In Fig. 17-19b, the left end of the
string is fastened to a light ring that
is free to slide without friclion along
a rod. When the incident pulse
arrives, the ring moves up the rod.
As the ring moves, it pulls on the
string producing a reflected pulse
with the same sign and amplitude as
the incident pulse.
reflected
wave
reflected
wave
Thus, in such a "soft" reflection, the
incident and reflected pulses
reinforce each other, creating an
antinode at the end of the string; the
maximum displacement is twice the
amplitude of either of these pulses.
17-12 Standing Waves and Resonance
Left
end
clamp
right
end
node
clamp
x
antinode
A string is stretched between two clamps. A continuous
sinusoidal wave is send toward the right. At the right end
it reflects back to the left end, and so on. The right-going
and the left-going waves overlapp each other and interfere
with each other.
For certain frequencies, the interference produces a standing wave pattern (or oscillation mode) with nodes and large
antinodes like those in the above figure.
17-12 Standing Waves and Resonance
Left
end
right
end
clamp
clamp
x
resonant frequencies
Such a standing wave is said to be produced at resonance,
and the string is said to resonate at these certain frequencies,
called resonant frequencies.
clamp
some other frequencies
clamp
If the string is oscillated at some flequency other than a
resonant frequency, a standing wave is not set up. Then the
interference of the right-going and left-going traveling waves
results in only small (perhapr imperceptible) oscillations of
the string.
17-12 Standing Waves and Resonance
To find expressions for the resonant frequencies of the
string, we note that a node
must exist at each of its ends,
because each end is fixed and
cannot oscillate.
Several simplest patterns:
L
1
2
L
22
2
L
33
2
 2  L ,   2L
one
loop
2 2  L,   L
two
loops
The simplest
pattern
Two nodes
one antinode
The second
simple pattern
three nodes
two antinodes
The third
simple pattern
four nodes
3
three antinodes
2
three
2  L ,   L loops
3
17-12 Standing Waves and Resonance
We could draw increasingly
more complicated patterns.
Thus, a standing wave can be
set up on a string of length L
by a wave with a wavelength
equal to one of the values
2L

n
n  1, 2,3
(17-52)
The resonant frequency
that correspond to these
wavelength
v
v
f  n
n  1,2,3
 2L
L
1
2
L
22
2
L
33
2
(17-52)
Here v is the speed of
traveling waves on the
string.
17-12 Standing Waves and Resonance
v
v
f  n
n  1,2,3
 2L
n  1,
(17-52)
L
f1  v 2L
fundamental made or
the first harmonic.
n  2,
f2  v L
the second harmonic.
n  3, f3  3 v 2L
the third harmonic.
The collection of all
possible oscillation
models is called the
harmonic series, and
n
is called the
harmonic number of
the nth harmonic
1
2
L
22
2
L
33
2
17-12 Standing Waves and Resonance
Sample Problem 17-7 P391
P
Q
m