Chapter 19-20 Mechanical Waves

Shock waves produced by the wings of a navy FA 18 jet.the sudden decrease in air pressure cause water molecules to condense,forming a fog.

How does a scorpion( 蝎子 ) detect a beetle( 甲虫 ) without using sight or sound?

An ultrasound( 超声波 ) image of a fetus( 胎儿 ) searching for a thumb to suck.The image of a transistor( 晶体管 ), viewed in an acoustic microscope( 声学显微镜 ).How does a bat detect( 发现 ) a moth( 蛾子 ) in total darkness?

Waves

Mechanical Waves

ElectromagneticWaves

Substantial Waves

The condition to form the wave

The condition to form the wave

Basic wavequantity

Basic wavequantity Wave equationWave equation

relationrelation

Interference conditionInterference condition

Mechanic waveMechanic wave

Energy in wave

Energy in wave

Interference of wave

Interference of wave

Constructive and destructivecondition

Constructive and destructivecondition

Description of waveDescription of wave

Standing waveStanding wave

Key Terms:mechanical wavemediumtransverse wavelongitudinal waveperiodic wave sinusoidal wavewave speedwave lengthwave functionwave equation

boundary condition principle of superposition interference node

Key Terms:

Antinode

standing wave

traveling wave

destructive interference

constructive interference

fundamental frequency

harmonics

harmonics series

overtone

normal mode

Harmonic content

resonance

resonance curve

1.1 conceptual idea on wave

1. wave equation

1) mechanic wave

a disturbance propagate in medium

2) classification of mechanic wave：transverse and longitudinal wave

caution：caution：

disturbance (wave origin)

medium　　

transverse wave longitudinal wave

The flip of the wrist pulse constitutes a disturbance that travels along the rope.

the particle in rope itself is not transported, only the pulse or status of oscillation is transport.

Caution:Caution:

B) Wavelength()The distance from one crest to the next, or

from one trough to the next.

A) Period(T)Time for one medium particle to complete a cycl

e.

caution:caution:

3) Basic quantities

Space Periodicity

Time Periodicity

period of wave is equal to the period of the oscillation source.

xO v

yA

Tell the difference of the speed of wave and speed of oscillation particle.

c) Wave Speed(v)

fT

v

Caution:Caution:

xO v

yA

The wave pattern travels with constant speed v and advances a distance of one wavelength in one period T.

The speed of particle:

change rate of its displacement with time t (in x-t graph,it’s slope)

the speed of wave:

the speed of the propagation phase or the speed of the disturbance status or the distance of the wave propagate in one unit time (decide by system)

y

t

y

x1 21 2

v

x

Speed of Transverse Wave on a String:

F: tension in the string

: mass per unit lenth (linear mass density)

Fv

Speed of Longitudinal Waves in a Fluid :

B: bulk modulus ( 体积弹性模量 )

: mass per unit volume ( 体密度）

Bv

Speed of Longitudinal Waves in a Solid Rod :

Y

v Y: Young’s modulus ( 扬氏模量 )

: mass per unit volume ( 体密度）

Wave Speed(v) The wave speed is determined by the mechanical properties of the medium.

Solution:

1311

1006.57800

102

smY

v

Speed of wave in the metal pipe

st 270.0)1006.5

100

345

100(

3

Example: A steel pipe 100m long is struck at one end. A person at the other end hears two sounds as a result of two longitudinal waves, one traveling in the metal pipe and the other traveling in the air. What is the time interval between the two sounds? Take Young’s modulus of steel to be 21011Pa, the density of steel to be 7800kg.m-s, and the speed of sound in air to be 345m.s-1.

Sound waves are longitudinal waves in air. The speed of sound depends on temperature; at 20oC it is 344m/s. what is wavelength of a sound wave in air at 20oC if the frequency is f=262Hz (the approximate frequency of middle C on a piano)?

vT .m31.1s262

s/m3441 f

v

Example:When a beetle moves along the sand within a few tens of centimeters of this sand scorpion, the scorpion immediately turns toward the beetle and dash to it. The scorpion can do this without seeing the beetle,how?

lt v

d

v

dt

Solution:

tl

lt

vv

vvtd

2. Mathematical Description of a Wave

A function that describes the position of any individual particles in the medium at any time during wave propagation.

)t,x(yy

2.1 Wave Function )t,r(yy

2.2 Wave Function for a Sinusoidal Wave

xxO

y v

P

xxO

y v

P

x: equilibrium position of a particle

y: displacement of the particle

Suppose at point O: )t,0x(yy How about the any other point, such as point P?

)cos( tA

The wave travels from O to P:

v

xxt 0P ])(cos[),(

v

xtAtxy

v

x

)t,x(yy

xxO

y v

P

If the wave travels in the negative x-axis, How to write the wave function?

Suppose at point O: )t,0x(yy )tcos(A

The wave travels from P to O:v

xOP

v

xOP

v

xt

])v

xt(cos[A)t,x(y

])v

xt(cos[A)t,x(y vT,

T

2

)x2

tcos(A)t,x(y

v

2k:assume

Wave Number)kxtcos(A)t,x(y

)x

T

t(2cosA)t,x(y

kxx2

xv

:"" The wave travels in +x-axis.:"" The wave travels in -x-axis.

A) four key factor

B) show the relation between the time and space

caution：

caution： ,,, Ak

])v

xt(cos[A)t,x(y

If t is determined ( t=t0):

])v

xt(cos[Ay 0

“y” describe the motion of a particle at x=x0 as a function of time.

v

)s(t

)cm(y

O

A

1

0x

)m(x

)cm(y

O

A

1

0t

])v

xt(cos[Ay 0

“y” describe the shape of the wave at t=t0

T

If x is determined ( x=x0):

If “x” and “t” are not determined :

tv])

v

xt(cos[A)t,x(y

])v

tvxtt(cos[A)t,x(y

)m(x

)cm(y

O

A

t vx

tt

C) the wave speed is the speed of a propagation of phase

Tkv

kdt

dx .conkxt

)kxtcos(A)t,x(y

)m(x

)cm(y

O

A

1

v 0t

)kxtsin(At

)t,x(yv y

Av ,maxy

)kxtcos(At

)t,x(ya 2

2

2

y

2,maxy Aa

)kxtcos(Akx

)t,x(y 22

2

yv

yv

yayv

ya 22

2

22

22

vkx)t,x(y

t)t,x(y

vk

2

2

22

2

t

)t,x(y

v

1

x

)t,x(y

D) Particle Velocity and Acceleration in a Sinusoidal Wave

E) the procedure to construct wave equation

2) Find the oscillation in any point x (wave equation)

1) Find the oscillation in a reference point xgiv.

)cos( tAy

Ogivx anyx

givany xxx

]tcos[Ay

If v is along +x, use – sign in the function;

If v is along –x, use + sign in the function

xv )( givxx

v

x2 )(

2givxx

Example: The equation of a transverse wave on string is as follows,

The tension in rope is 15N, what is the wave speed and linear density of the string

))(60020sin(2 SItxy

Solution: )cos( xv

tAy

)2

20600cos(2

xty

smv /30,600

mkgv

FFv /0167.0,

2

Example: A transverse sine wave of amplitude 0.10m and wavelength 2m travels from left to right along a long horizontal stretched string with a speed of 1m.s-1. Take the origin at the left end of the undisturbed string. At time t=0, the left end of the string is at the origin and is moving downward.a) What is the equation of the wave?b) What is the equation of motion of the left end of the string?c) What is the equation of motion of a particle 1.5m to the right of the origin?d) What is the maximum magnitude of transverse velocity of any particle of the string?e) Find the transverse displacement and the transverse velocity of a particle 1.5 m to the right of the origin, at time t=3.25 s.

Draw a circle of reference for Origin at t=0

x1.0 1.0O

2

)2xtcos(1.0y

)xtcos(1.0y

a)

)xv

tcos(Ayassume 1sm1v,m2,m1.0A:known

T

2,s2

vT

)xtcos(1.0y

Solution:

b)c)

)2tcos(1.0y,0x ,m5.1x )223tcos(1.0y

tcos1.0

d)

x)t

y(v

)2xtsin(1.0

s/m314.01.0vmax e)

)2xtcos(1.0y

m)25.125.3cos(1.0y m0707.0

)25.125.3sin(1.0v s/m222.0

x5cmA B

u

Example: a wave propagate along negative x direction

with u=20cm/s ， yA=0.4cos4t(cm), find wave equatio

n ：(1)assume the original point is at A ； (2)the original point at B

,/20,4)1 scmu

)

20

5(4cos)2 tAyb

)

54cos4.0)

5(4cos x

tu

x

utAy

Solution:

cmx

ty

204cos4.0

)s(t

)cm(yP

O

2

1.0 2.0

)m(x

)cm(y

O

2

0.2

0t

P Q

Example: Known: the graph of y(x,t) versus x at t=0, the graph of y(x,t) versus t at the point P. a) What is the function of the wave? b) What the velocity of point Q at t=0.1s.

v

cm)2t10cos(2y

s2.0T,m0.2,cm2A

cm)2t10cos(2y

:assum

/)0.1x(2

cm]2

xt10cos[2y

Solution: From the graph:

Motion of point P:

3. energy in wave motion

3.1 Energy and Average Power

)cos(),( kxtAtxy pk EEE

1) kinetic energy

)(sin21)(

21

21 22222 kxtAdx

ty

dxdmdEk v

x x+dxx

y(x,t) y(x+dx,t)

2) potential energy

It can be provepk EE

kxtAdx

EEE pk

222 sin

3.2　 Intensity of the wave

22 AI 2) Only valid for mechanic wave.for electronic waves, the intensity is independent of the value of

Caution:Caution:

)sin(2 22 kxtAvdt

dE

dt

dEP k

2222

0 2

1

2

11AFAvpdt

Tp

T

av

22

2

1ABI

22

2

1AYI

For fluids in pipe

For solid rod

1)

What happens when two waves meet while they travel through the same medium?

4. Wave Interference and Normal Modes

4.1 principle of superposition:

)t,x(y)t,x(yy 21 2

2

22

2

t

)t,x(y

v

1

x

)t,x(y

The wave equation is linear.

If any two functions y1(x,t) and y2(x,t) satisfy the wave equation separately, their sum y1(x,t) + y2(x,t) also satisfies it and is therefore a physically possible motion.

4.2 Phenomena of interference

1S 1r

2r2S

P

);tcos(A)t,S(y 101110

)tcos(A)t,S(y 202220 Suppose:

then: )r2

tcos(A)t,P(y 11011

)r2

tcos(A)t,P(y 22022

Overlap at point P:

)tcos(Ayyy 21

4.3 Conditions of Interference

(a) Two waves must have the same frequency.(b) The same oscillation direction.(c) Constant phase difference .

4.4 Quantitative Analysis

)rr(2

)( 121020

cosII2IIAI 21212

)rr(2

)( 121020

,k2

,)1k2(

constructive

destructive

)rr(2

:then,0if 121020

12 rrk

2)1k2(

constructive

destructive

cosAA2AAA 2122

21

2

)tcos(Ayyy 21

1S 1r

2r2S

P

)rr(2

)( 121020

,k2

,)1k2(

constructive

destructive

Example: Two loudspeakers,separated by a distance of 2m are in phase. Assume the amplitude of a sound from the speakers are approximately the same at the position of a listener who is 3.75m directly in front of one speakers. For what frequencies in the audible range(20 hz to 20khz) does the listener hear a minimum signal?

the path difference:

Example: Two loudspeakers, A and B, are driver by the same amplifier and emit pure sinusoidal waves in phase. If the speed of sound is 350m/s, a) for what frequencies does constructive interference occur at point P; b) for what frequencies does destructive interference occur at point P?

P

A Bm00.2 m00.1

m00.4

Solution:

BPAPr m35.012.447.4

krwhen Constructive interference occur:

f/v

f/kv ...2,1k r/kvf Hz)k1000(

Destructive interference occur:2/)1k2(rwhen ...2,1k

r/v)2/1k(f Hz)500k1000( See page643, 20-21, 20-22

Example: A sound with a 40cm wavelength travels rightward from a source and through a tube that consists of a straight portion and a half -circle.Part of the sound wave travels through the half circle and then rejoins the rest of the wave, which goes directly through the straight portion. This rejoining result is interference.what is the smallest radius r that results in an intensity minimum at the detector.

Standing Waves

1.condition：

2.mathematics formula

Suppose two interference wave

)2cos(),(1 xtAtxy )2cos(),(2

xtAtxy

The resultant wave

txAtxytxytxy cos2cos2),(),(),( 21

Two interference waves equals in amplitude, and moves in opposite direction.

4.5 Standing Wave

mx 2

2mx

，

AA 2 antinodes

nodes )12(2 mx

4)12( mx0A

2,1,0 m

22 mx

Distance between nodes or antinodes

the wave pattern remains in the same position, and its amplitude fluctuates, vary from the position of the particle, no relation with time t.

xAA 2cos2

(1) amplitude

discussion：discussion：

txAtxytxytxy cos2cos2),(),(),( 21

02cos x t

02cos x t

Particle between any successive pair of nodes oscillate in

phase, and particle beside each node oscillate out of phase

(2) phase

02cos x X is a turning point

2

2

2

x

2

32

2

x

(3) the energy in standing wave:

no energy transfer

A standing wave does not transfer energy from one end to the other. There is a local flow of energy from each node to the adjacent antinodes and back, but the average rate of energy transfer is zero at every point.

Caution:Caution:

The above conclusion is in contrast to the phase differences between oscillations of adjacent points in traveling wave.

Tv,...3,2,1n,

L2

vnfn

1n nfL2

vnf Harmonic series

These special frequencies and their associated wave patterns are called normal modes

L2

vf1 Fundamental frequency

4.5 Normal Modes of a String

If a string of length L, rigidly held at both ends, then L must satisfied:

,...3,2,1n,2

nL n

,...3,2,1n,n

L2n

21L

Guitar Strings

Example:A nylon guitar string has a linear density of 7.2 g/m and is under a tension of 150N, the fixed supports are 90cm apart. The string is oscillating in the standing wave pattern shown in the following figure. Calculate speed , wavelength and frequency of the traveling wave whose superposition gives this standing wave.

Doppler Effect

Doppler effect occurs for all types of waves, water, sound and even light. This is how we know the universe is expanding.

Doppler EffectDoppler Effect

Blue Shift – Source (star) is moving toward earthRed Shift - Source (star) is moving away from earthLead to Big Bang Theory

Doppler Effect – frequency shift that is the result of relative motion between the source of waves and an observer

2. stationary source, and moving listener

ff

f llll v

vv

v

vvvv

vR

1. stationary source and stationary listener

λf v

ff ll v

vv

fvv

v

TvvT

vvf

ssl

fvv

vf

sl

3　 stationary listener and moving sources

4. Both listener and source move

Move toward each other ffS

Rl vv

vv

Away from each other ffS

Rl vv

vv

f

vvTvvTv s

ss

)(0

0

Example:A train whistle at rest has a frequency of 3000 Hertz. If you are standing still and observe the frequency to be 3010 Hertz, then you can conclude that...

a) the train is moving away from you.b) the train is moving toward youc) not enough information is given

A French submarine and a U.S submarine move toward each other during maneuvers in motionless water in North Atlantic. The French sub moves at 50km/h, and the U.S sub at 70km/h. The French sub sends out sonar signal at 1000hz, sonar wave travel at 5470km/h. What is the signals frequency as detected by U.S sub. What frequency is detected by French sub in the signal reflected back to it by the U.S sub?

Shock Waves

The source is moving faster than the wave speed in the medium.

A shock wave is formed and it is very difficult to break through the previous wave barrier.

These waves produce sonic booms.

ffS

Rl vv

vv

vvs