Doppler effect, moving sources/receivers

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Physics 207: Lecture 29, Pg 1 Lecture 29 Goals: Goals:

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Chapter 20Chapter 20

Work with a few important characteristics of sound waves. (e.g., Doppler effect)

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Chapter 21Chapter 21

Recognize standing waves are the superposition of two traveling waves of same frequency

Study the basic properties of standing waves

Model interference occurs in one and two dimensions

Understand beats as the superposition of two waves of unequal frequency.

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AssignmentAssignment

HW12, Due Friday, May 8th

Thursday, Finish up, begin review for final, evaluations

Physics 207: Lecture 29, Pg 2

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Doppler effect, moving sources/receivers

If the source of sound is moving

Toward the observer ⇒ λseems smaller

Away from observer ⇒ λseems larger

If the observer is moving

Toward the source ⇒ λseems smaller

Away from source ⇒ λseems larger v v_s

f

−

=

1

source observer source o observer

v

1

f













+

=

Doppler Example Audio Doppler Example Visual

v

_s

f

+

=

1

source observer source o observer

v

1

f













−

=

Doppler Example

A speaker sits on a small moving cart and emits a short 1

Watt sine wave pulse at 340 Hz (the speed of sound in air is 340 m/s, so λ= 1m ). The cart is 30 meters away from the wall and moving towards it at 20 m/s.

The sound reflects perfectly from the wall. To an observer

on the cart, what is the Doppler shifted frequency of the directly reflected sound?

Considering only the position of the cart, what is the

intensity of the reflected sound? (In principle on would have to look at the energy per unit time in the moving frame.)

t₀

A

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Doppler Example

The sound reflects perfectly from the wall. To an observer on

the cart, what is the Doppler shifted frequency of the directly reflected sound?

At the wall: f_wall = 340 / (1-20/340) = 361 Hz

Wall becomes “source” for the subsequent part

At the speaker f _’ = f_wall(1+ 20/340) = 382 Hz

t₀ 30 m t1 v v_s

f

−

=

1

source observer source o observer v v 1 f f       + = Physics 207: Lecture 29, Pg 6

Example Interference

Considering only the position of the cart, what is the intensity of

the reflected sound to this observer? (In principle one would have to look at the energy per unit time in the moving frame.)

v_cart∆t + v_sound∆t = 2 x 30 m = 60 m ∆t = 60 / (340+20) = 0.17 s dsound= 340 * 0.17 m = 58 m I = 1 / (4π 582_{) = 2.4 x 10}-5_W/m2 _{or 74 dBs} t₀ 30 m t1

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Doppler effect, moving sources/receivers

Three key pieces of information Time of echo

Intensity of echo Frequency of echo

Plus prior knowledge of object being studied

With modern technology (analog and digital) this can be done in real time.

Superposition Q: What happens when two waves “collide” ? A: They ADD together!

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Interference of Waves

2D Surface Waves on Water

In phase sources separated by a distance d

d

Principle of superposition

The superposition of 2 or more waves is called interference

Constructive interference: These two waves are in phase. Their crests are aligned.

Their superposition produces a wave with amplitude 2a

Destructive interference: These two waves are out of

phase.

The crests of one are aligned with the troughs of the other.

Their superposition produces a wave with zero amplitude

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Interference: space and time

Is this a point of constructive or destructive interference?

What do we need to do to make the sound from these two speakers interfere constructively?

Interference of Sound

Sound waves interfere, just like transverse waves do. The

resulting wave (displacement, pressure) is the sum of the two (or more) waves you started with.

|

r

₁

r

₂

r

=

r

−

r

∆

,... 2 , 1 , 0 ) 2 1 ( 2 2 ce interferen e destructiv Maximum ) ( 2 2 2 2 ce interferen ve constructi Maximum 2 1 2 1 2 1 = + = − + ∆ = ∆ = − + ∆ = ∆ = − + ∆ = ∆ m m r m r m r

π

φ

λ

π

φ

λ

φ

π

λ

φ

π

λ

π

φ

λ

π

φ

] ) / / ( 2 cos[ ) , ( ₂ ₂ 2 2 =

π

r

λ

−t T +

φ

r A t r D

]

)

/

(

2

cos[

)

,

(

₁ ₁ 1 1

=

π

r

λ

−

t

T

+

φ

r

A

t

r

D

r

∆

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Example Interference

A speaker sits on a pedestal 2 m tall and emits a sine wave

at 343 Hz (the speed of sound in air is 343 m/s, so λ= 1m ). Only the direct sound wave and that which reflects off the ground at a position half-way between the speaker and the person (also 2 m tall) makes it to the persons ear.

How close to the speaker can the person stand (A to D) so

they hear a maximum sound intensity assuming there is no phase change at the ground (this is a bad assumption)?

The distances AD and BCD have equal transit times so the sound waves will be in phase. The only need is for AB = λ

t₁ t₀ t₀ A B A D C d h Physics 207: Lecture 29, Pg 14

Example Interference

The geometry dictates everything else.

AB = λ AD = BC+CD = BC + (h2_{+ (d/2)}2₎½_{= d} AC = AB+BC = λ +BC = (h2_{+ d/2}2₎½ Eliminating BC gives λ+d = 2 (h2_{+ d}2_/4)½ λ + 2λd + d2_{= 4 h}2_{+ d}2 1 + 2d = 4 h2 _/λ _{d = 2 h}2 _/λ _{– ½} = 7.5 m t₁ t₀ t₀ A B A D C 7.5 4.25 3.25

Because the ground is more dense than air there will be a phase change of πand so we really should set AB toλ/2 or 0.5 m.

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Exercise Superposition

Two continuous harmonic waves with the same frequency

and amplitude but, at a certain time, have a phase

difference of 170°are superimposed. Which of the follo wing best represents the resultant wave at this moment?

(A) (E) (D) (C) (B) Original wave

(the other has a different phase)

Wave motion at interfaces Reflection of a Wave, Fixed End When the pulse reaches the support,

the pulse moves back along the string in the opposite direction

This is the reflection of the pulse

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Reflection of a Wave, Fixed End

Animation

Reflection of a Wave, Free End

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Transmission of a Wave, Case 1

When the boundary is intermediate between the last two extremes ( The right hand rope is massive or massless.) then part of the energy in the incident pulse is reflected and part is transmitted

Some energy passes through the boundary Here µ_rhs>µ_lhs

Animation

Transmission of a Wave, Case 2

Now assume a heavier string is attached to a light string

Part of the pulse is reflected and part is transmitted The reflected part is not inverted

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Standing waves

Two waves traveling in opposite direction interfere with each other.

If the conditions are right, same k & ω, their interference generates a standing wave:

D_Right(x,t)= a sin(kx-ωt) D_Left(x,t)= a sin(kx+ωt)

A standing wave does not propagate in space, it “stands” in place. A standing wave has nodes and antinodes

D(x,t)= D_L(x,t) + D_R(x,t) D(x,t)= 2a sin(kx) cos(ωt) The outer curve is the

amplitude function A(x) = 2a sin(kx) when ωt = 2πn n = 0,1,2,… k = wave number = 2 / Nodes Anti-nodes Physics 207: Lecture 29, Pg 22

Standing waves on a string

Longest wavelength allowed is one half of a wave

Fundamental: λ/2 = L λ = 2 L

,...

3

,

2

,

1

2

=

m

f

v

m

L

m m

λ

Recall v = f

λ

L

v

m

f

_m

2

=

Overtones m > 1

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Vibrating Strings- Superposition Principle

Violin, viola, cello, string bass Guitars Ukuleles Mandolins Banjos D(x,0) A n ti n o d e D (0 ,t ) Physics 207: Lecture 29, Pg 24

Standing waves in a pipe

Open end: Must be a displacement antinode (pressure minimum) Closed end: Must be a displacement node (pressure maximum) Blue curves are displacement oscillations. Red curves, pressure.

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Standing waves in a pipe

,...

3

,

2

,

1

2

=

m

L

v

m

f

m

L

m m

λ

,...

3

,

2

,

1

2

=

m

L

v

m

f

m

L

m m

λ

,...

5

,

3

,

1

4

=

m

L

v

m

f

m

L

m m

λ

Combining Waves

Fourier Synthesis

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Physics 207: Lecture 29, Pg 27 Lecture 29

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AssignmentAssignment