Chapter 18Wave Motion

18-1 Mechanical wavesIn this chapter, we consider only mechanical waves, such as sound waves, water waves, and the waves transmitting in a guitar’s strings.

• Elastic mediums are needed for the travel of mechanical waves.

• Mechanical waves can appear when an initial disturbance is made to the mediums.

On a microscopic level, the forces between atoms in the mediums are responsible for the propagation of the waves.

•The particles of the medium do not experience any net displacement in the direction of the wave-as the wave passes, the particles simply move back and forth through small distance about their equilibrium position.

What is a wave?It is the process of propagating oscillation in space.

What are transmitted by a wave?Energy, momentum, phase…, but the particles are not.

18-2 Types of waves

Waves can be classified according to their properties as following.

1.According to direction of particle motion(a)“Transverse waves(横波 )”: If the motion of the particle is perpendicular to the direction of propagation of the waves itself.

(b)“Longitudinal wave(纵波 )”: If the motion of the particle is parallel to the direction of propagation of the waves.

See动画库 \波动与光学夹 \2-01 波的产生 2 3

2. According to number of dimensions

1-D Waves moving along the string or spring 2-D Surface waves or ripple on water3-D Waves traveling radially outward from a small source, such as sound waves and light waves.

3 According to periodicity

pulse waves or periodic wave.

The simplest periodic wave is a “simple harmonic wave’’ in which each particle undergoes simple harmonic motion.

y

xo

The simplest periodic wave

Other kinds of periodic waves:

Square waveTriangle wavemodulated waveSawtoothed wave

4. According to shape of wavefronts

(a) The definitions of ‘wave surface’ (波面或同相面 )and ‘wavefront’(波前或波阵面 )?

See动画库 \波动与光学夹 \2-02 波的描述 1

(b) The definition of ‘a ray’(波线 ): A line normal to the wavefronts, indicating the direction of motion of the waves.

Wavefronts are always direction of Ray

Plane wave: The wavefronts are planes, and the

rays are parallel straight lines.

Spherical wave: The wavefronts are spherical,

and the rays are radial lines leaving the point

source in all directions.

★ Two different types of wavefronts: Plane waves Spherical waves

Ray(波线 ) Wave surface(波面 ) Wavefront(波前 )

*

Spherical wave Plane wave

波前

波面

ray

5. Waves in different fields in physics

sound waveswater wavesearthquake waveslight waveselectromagnetic wavesgravitational wavesmatter waveslattice waves

18-3 Traveling waves(行波 )• All the waves would travel or propagate, why here say ‘traveling waves’?

(with respect to ‘standing wave’(驻波 ))

• Definition of traveling waves: The waves formed and traveling in an open medium system.

• Description of traveling waves We use a 1-D simple harmonic, transverse, plane wave as an example

• Mathematics expressions The vibration displacement y as a function of t and x.

The difference between vibration and wave motion: Vibration y(t): displacement as a function of time Wave y(x,t): displacement as a function of both time and distance

)2

sin()0,( xyxy m

Fig 18-6

vt

y

x

t = 0 t = t

υ

What we want to know: ),( txy ))(

2sin( vtxym

1. Equation of a sine wave

If there is initial phase constant in the sinusoidal waves, the general equation of the wave at time t is:

))(2

sin(),(

vtxytxy m (18-16)

Several important concepts about waves:1) The period T of the wave is the time necessary forpoint at any particular x coordinate to undergo one complete cycle of transverse motion. During this time T, the wave travels a distance that must correspond to one wavelength .

vT

2) The wavelength : the length of a complete wave shape.

3) The frequency of the wave : T

f1

4) The wave number:2

k

5) The angular frequency : fT

22

(18-16) )sin(),( tkxytxy m

Note that:

speed of the wave

The equation of a sine wave traveling in direction is

x

)sin(),( tkxytxy m

The equation of a sine wave traveling in the direction is

x

)sin(),( tkxytxy m

(18-11)

(18-12)

kfv

(18-13)

(18-16)

2. Transverse velocity of a particle

vNote that is the speed of wave transmitting.

What is the velocity of particle oscillating?

---- It is called transverse velocity of a particle for transverse wave

)cos(

)]sin([),(

tkxy

tkxytt

ytxu

m

my

Transverse velocity:

Tansverse acceleration:

ytkxydt

ydtxa my

222

2

)sin(),(

(18-14)

(18-15)

3. Phase and phase constant

)sin(),( tkxytxy m

)( tkx

If the equation of the wave is:

Phase

phase constant

Eq(18-16) can be written in two equivalent forms:

(18-17a)

(18-17b)

])(sin[),( tk

xkytxy m

)](sin[),( tkxytxy m

(18-16)

In y-x, wave A is ahead of wave B by a distance /k

In y-t, wave A is ahead of wave B by a time /ω

(a)

x

k

B A

y

y = ymsin(kx – ωt – )Two waves A and B:

y = ymsin(kx – ωt ) wave A wave B

(b)

t

Fig 18-7

y

A B

])(sin[),( tk

xkytxy m

)](sin[),( tkxytxy m

lead lag

Sample problem 18-1

A transverse sinusoidal wave is generated at one end of a long horizontal string by a bar that movesthe end up and down through a distance of 1.30cm. The motion is repeated regularly 125 times per second(a) If the distance between adjacent wave crests is 15.6 cm, find the amplitude, frequency, speed, and wavelength of the wave .(b) Assuming the wave moves in the +x direction and that at t=0, the element of the string at x=0 is at its equilibrium position y=0 and moving downward, find the equation of the wave.

Solution:

(a) The amplitude

frequency

wavelength

speed

(b) The general expression for a sinusoidal waves

is given by Eq(18-16)

cmcmym 65.02

30.1

Hzf 125

smfv /5.19cm6.15

)sin(),( tkxytxy m

Imposing the given initial condition ( and

for x=0 and t=0 ) yields

and

thus ,

0)sin( my 0cos my

0

])/786()/3.40sin[()65.0(

)sin(),(

tsradxmradcm

tkxytxy m

0y 0

t

y

Sample problem 18-2

In sample problem 18-1.

(a) Find expressions for the velocity and acceleration of a particle P at

(b) Evaluate the y, , of this particle at

Solution:

(a)

(b)

mxP 245.0

ya mst 0.15

ytxa py2),(

cmy 61.0scmu y /173

25 /108.3 scma y

)cos(),( tkxytxu mpy

yu

18-4 Wave velocity (speed)

1) Phase velocity

• Definition: The velocity of the motion of certain phase in a wave (for monochromatic wave(单色波 ,单一频率的波 ))

k

ωv

• Wave speed on a stretched string

Phase velocity vs group velocity

From dimensional analysis:F

v

From mechanical analysis:F

v

: source of the wave: the medium (non-dispersive)

v

oR

l

F

F

R

lFFFF ynet

2sin2,

0, xnetF

R

vl

R

vmma

R

lF y

22

F

v (18-19)

F --- tension force exerted between neighboring elementsμ --- mass density (mass/unit length)

v

v

Wave velocity

• When a wave passes from one medium to another medium, the frequency keeps the same, namely

21 ff may vary. λ and v

2) Group velocity

For a group of waves with different :In non- dispersive medium,

All the waves with different moves with same speed.

time = 0

time = tx

x

υ

Shape keeps

, determined only by the mediumv

time 0

time tx

x

In dispersive medium,

All the waves with different moves with different speeds.

Shape does notkeeps!!!

Group speedGroup speed is needed to describe the waves.

dk

dωv

In this chapter, all the mediums met is assumed to be nondispersive.

A crazy physicist!?

18-5* The wave equation

18-6 Energy in wave motion

1. Energy in wave

motionFig18-11a shows a

wave traveling along the

string at times and

( a time later ). 1t 2t

4

T

A B

y

x

(a)

dx

dydl (b)

yu

1t 2t

yu

time time

Fig 18-11

Wave transmits energy.

What do we want to calculate?

• dK/dt – the rate at which kinetic energy is transportedby wave.

• dU/dt – the rate at which potential energy is transported.

22 )]cos()[(2

1

2

1tkxydxudmdK my

For : dK/dt

)(cos2

1 222 tkxdt

dxy

dt

dKm

v)(cos

2

1 222 tkxvydt

dKm (18 - 26)

)(cos2

1 222 tkxdxym

For :dU/dt )( dxdlFdU

]1)(1[])()([ 222 dx

dyFdxdxdydxFdu

22 )()( dydxdl

The quantity is the slope of the string, and if the amplitude of the wave is not too large this slope will be small.

dx

dy

z2

11z)(1 using 1/2

22 )(2

1]1)(

2

11[

x

yFdx

x

yFdxdu

;)/( 22 kvF

)cos( tkxkyx

ym

vdt

dx

dt

dKtkxvy

dt

dUm )(cos

2

1 222 (18-29)

Note that:

(a) dK and dU are both zero when the element has its maximum displacement ( the element at relaxed length ).

(b) The mechanical energy is not constant, because the mass element is not an isolated

system—neighboring mass elements are doing work on it to change its energy.

dKdUdE

)(cos2

1 222 tkxdxydUdK m

2. Power (功率 ) and intensity（能流密度）

• Power： the rate at which mechanical energy is transmitted.

dt

dEP

dKdKdUdE 2

)(cos222 tkxvym (18-30)

Average power :avP

T

av dtdt

dE

TP

0

1vym

22

2

1 (18-32)

• Intensity I:A

PI av (18-33)

For spherical wave: ;1

4 22 rr

PI av

rym

1

18-7 The principle of superposition The principle of superposition:

Two or more waves travel simultaneously through the same region of space, the superposition principle holds.

...),(),(),( 21 txytxytxy (18-34)

See动画库 \波动与光学夹 \2-03 波的叠加原理

18-8 Interference(干涉 ) of waves

When two or more waves combine at a particular point, they are said to “interfere”, and the phenomenon is called “interference.”

We consider a general case, the equation of the two waves are

Using the principle of superposition,

)sin(),( 11 tkxytxy m

)sin(),( 22 tkxytxy m

)sin()]2/cos(2[

)]sin()[sin(

),(),(),(

'

21

21

tkxy

tkxtkxy

txytxytxy

m

m

(18-36)

(18-37)

where ,

This resultant wave corresponds to a new wave having the same frequency but with an amplitude

1. If (in phase(同相 )) , the resultant amplitude

is , this case is known as constructive Interference (相长干涉 ).

2. If (out of phase (反相 ) ), the resultant amplitude is nearly zero, this is destructive interference (相消干涉 ).

12 2

12'

.)2/cos(2 my

0

my2

180

The resultant amplitude is shown in Fig18-16.

x x

21 yy 1y

2y

21 yy 1y

2y

0 180

Fig 18-16

波源发出的波，到达两个狭缝时，成为两列频率相同、振动频率相同、振动方向平行、相位方向平行、相位相同或相位差恒相同或相位差恒定定的波，在狭缝后面的屏幕上产生 波 的 干 涉 现象。呈现明暗相间的条纹。

Interference of Waves

Young’s double slit light-interference experimentRanked as 5 in top 10 beautiful experiments in Physics

See动画库 \波动与光学夹 \2-04 波的干涉

One paradox (佯谬 ) about energy of wave interference:

两个沿相同方向传播的一维简谐波，它们的频率和振幅 A均相同。如果位相相反，那末叠加后振幅为零，波的能量哪里去了？

如果位相相同，叠加后振幅为 2A，在其它参数相同的情况下，波的能量正比于振幅的平方，两个波在叠加前能量为 A2 + A2，叠加后变为 (2A)2，能量怎么会多出来了？

vym22

2

1 avP

18-9 Standing waves

In previous section, we consider the effect of superposing two component waves of equal amplitude and frequency moving in the same direction on a string. What is the effect if the waves are moving along the string in opposite direction?

1. We represent the two waves by

)sin(1 tkxyy m

)sin(2 tkxyy m

Hence the resultant wave is:

(a) Eq(18-42) is the equation of a standing wave.

It is not a traveling wave, because x and t do not appear in the combination or , required for a traveling wave.

)sin()sin(21

tkxytkxy

yyy

mm

tkxyy m cos]sin2[

vtx vtx

(18-41)

(18-42) or

(b) Nodes (波节 ) and antinodes(波腹 ) of standing waves

In a standing wave, the amplitude is not the same for different particles. The behavior is different from that of a traveling wave.

Antinodes(波腹 )

The positions where the amplitude has a maximum value.

,2sin2 mm ykxy ,)2

1( nkxif n=0,1,2,…….

2)

2

1(

nxor (18-43)

...4

9,

4

7,

4

5,

4

3,

4

x

Nodes(波节 )The positions where the amplitude has a minimum value of zero.

,0sin2 kxym ,nkx if

or2

nx ,...2,

2

3,,

2,0

x

n=0,1,2,…,

See动画库 \波动与光学夹 \2-14驻波演示

To form a standing wave

n n n

aa

—— Forward wave

—— Backward wave—— Resultant wave

(c) Energy of standing waves

For standing waves, the energy can not be transported along it, because the energy cannot flow past the nodes, which are permanently at rest.

U k

Fig 18-18

U k U k U k

2. Reflection at a boundary

Let us discuss the case when a transverse pulse wave travels along a string and reaches an end (boundary).

What will happen when it is reflected at the boundary?

(a) If the reflection end is a fixed on, the reflected pulse is inverted (changes a phase of 180o), loses half wave at the boundary.

(b) If the reflection end is a free one, the reflected pulse is unchanged, no half wave loss at the boundary.

Suppose a pulse travels along a string and reaches an end

(a) (b)

(a) Reflection from a fixed end, a transverse wave undergoes a phase change of 180o

(b) At a free end, a transverse wave is reflected without change of phase.

See动画库 \波动与光学夹 \2-05半波损失

Fig 18-19

18-10 Standing waves and Resonance

L22

L

L

24

λL

2

3L

(a)

(b)

(c)

(d)

n=1

n=2

n=3

n=4Fig 18-20

...3,2,1,2

nnL

1) Standing waves in a string fixed at both ends

Thus the condition for a standing wave to be set up in a string of length L fixed at both ends is (18-45)

(18-46)

is the nth wavelength in this infinite series.

n is the number of half-wavelengths in the patterns.

is the frequency of the allowed standing waves,

(natural frequencies).

n

Ln

2

nL

vn

vf

nn 2

nf

2) Resonance in the stretched string

(a) In Fig18-20, a student begins to shake the string. If the frequency of the driving force matches one of the natural frequencies, we get a resonance in the string.

(b) If the student shakes the string at a frequency

that differs from one of the natural frequencies, the

reflected wave returns to the student’s hand out of

phase with the motion of the hand. No fixed standing

wave pattern is produced.

Sample problem 18-4

In Fig18-23, a motor sets the

string into motion at a

f=120Hz. The string has a

length of L=1.2m, and its

linear mass density .

Find the tension F, at which

we obtain the pattern of

motion having four loops?

mg /6.1 m

motor

Fig 18-23

Solution:

Substituting Eq(18-19) into Eq(18-46), we obtain

NmkgHzm

n

fLvF n

3.84

)/0016.0()120()2.1(4

4

2

22

2

222

L

vn

vf

nn 2

Sample problem 18-5

A violin string tuned to concert A (440Hz) has a

length L=0.34m. (a)What are the three longest

wavelengths of the resonances of the string? (b)

What are the corresponding wavelengths that reach

the ear of listener?

Solution:

(a)

(b)

,68.012 mLλ1 ,34.02

22 mL mL 23.03

23

n

Ln

2

string1n

stringn nf2L

vn

λ

vf ,, stringn,airairn, /fvλ

m/svair 343

Cover page of our text book

A Circular Quantum Corral constructed by 48 Fe atoms on Cu(111) at 4K in 1993.

Average diameter of ring = 14.26 nm.

Standing wave formed by electron wave interference inside a Quantum Corral (量子围栏 )

Double-walled ring of Fe atoms on Cu(111)

Quantum stadium

Schematic illustration of the process for sliding an atom across a surface.

Nature 344, 524 (1990)

Atomic-scale IBM logo produced by 35 Xe atoms on Ni(110) using scanning tunneling microscope (STM) at 4K in 1990.

Each letter is 5 nm from top to bottom.