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10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 1

PHY 113 C General Physics I11 AM -12:15 PM TR Olin 101

Plan for Lecture 18:Chapter 17 – Sound Waves, Doppler EffectChapter 18 – Superposition, Standing Waves,

Interference

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 2

10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 3

Reminder!

The linear wave equation

Wave variable

2 22

2 2

y yv t x

),( txyposition time

0x t λy(x, t) y sin 2π φ v = λ T T

Periodic traveling waves are solutions of the Wave Equation

Amplitude

wave length (m)period T = 1/f (s)

Solutions are of the form:

phase factor (radians)

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10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 4

SOUND WAVES obey the

**We’re neglecting attenuation. Attenuation is very important in ultrasonic imaging..

2 22

2 2

y yv t x

0xy(x, t) y sin 2 ft ; velocity of the wave: v = f

Periodic sound waves, general form

Amplitude (determines intensity or loudness)

wave length (m)frequency f (s-1) determines pitch

phase factor (radians)

linear wave equation**

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 5

SOUND: longitudinal mechanical vibrations transmitted through an elastic solid or a liquid or gas, with frequencies in the approximate range of 20 to 20,000 hertz. How are sound waves generated? What is the velocity of sound waves in air?

A simple model with the essential physics (for 113)

cylinder

piston gas

STEP ONE

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Questions:what has moved in the gas?in what direction have gas molecules moved? why isn’t the gas uniformly compressed ?

STEP TWO

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10/31/2013 PHY 113 C Fall 2013 -- Lecture 187

Notice that the velocity of the compressed zone is not the same as the velocity of the piston.

What happens if the piston oscillates at frequency f? Use animation……….

STEP THREE

1702 piston oscillation

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 8

A sound wave may be considered asa displacement wave :

Or

a pressure wave:

max s s cos(kx - t) nodes at / 2, 3 / 2,...

node

Antinode

What is y(x,t) in the wave equation of the sound wave?

max

max

P P sin(kx t) nodes at 0, , 2 ,...Since density is proportional to pressure,

sin(kx t)

antinode

node

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 9

The velocity of sound is easy to measure. v = 343 m/s at 20ºC in air.

How to relate the velocity of sound to the known properties of gases?Newton (1643-1727) worked out an analytic model, assuming isothermal compressions:

v = 298 m/s Laplace(1749-1827) pointed out that on the timescale of sound frequencies, compressions are not isothermal, they’re adiabatic (isentropic)

* * These fo

**

2 2 p2 2 02 2 0 v

If we use the linearized fluid equations in the ideal gas approximation,we find that the density fluctuations obey:

cPv 0 ; v 343m / s; here is the ratio .ct x

llow from conservation of mass, momentum, and energy.

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Speed of Sound in a Gas•Consider an element of the gas between the piston and the dashed line.•Initially, this element is in equilibrium under the influence of forces of equal magnitude.

– There is a force from the piston on left.

– There is another force from the rest of the gas.

– These forces have equal magnitudes of PA.

• P is the pressure of the gas.• A is the cross-sectional area of the

tube.

Section 17.2

Speed of Sound in a Gas, cont.•After a time period, Δt, the piston has moved to the right at a constant speed vx.

•The force has increased from PA to (P+ΔP)A.•The gas to the right of the element is undisturbed since the sound wave has not reached it yet.

Section 17.2

Impulse and Momentum, cont.•The change in momentum of the element of gas of mass m is

•Setting the impulse side of the equation equal to the momentum side and simplifying, the speed of sound in a gas becomes.

– The bulk modulus of the material is B.– The density of the material is

•

xm pvv A t ˆ p v i

Bv

Section 17.2

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10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 13

Intensity of sound waves;Powerdecibels (dB)

Velocity of sound:

Recall:For a string,

So, for sound,

tension elastic propertyvmass per unit length inertial property

B Bulk modulus elastic propertyvdensity inertial property

Definition of B: stress = modulus x strainstress = ; strain = p V / V

PB , definition of bulk modulusV V

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SUPERPOSITION of WAVES ( Chap. 18)

So far, we’ve considered only one travelling wave. However, a sum of two or more travelling waves will also satisfy the wave equation because the wave equation is linearThe Superposition Principle: If waves y1(x,t) and y2(x,t) are traveling through the same medium at the same time, the resultant wave function y3(x,t) = y1(x,t) + y2(x,t).

(This is not always true! If y1 or y2 become large enough, response of medium becomes nonlinear and the superposition principle is violated). Examples: music amplifiers; lasers.)

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 151803 superposition of two sinusoids

CASE 1. Superposition of 2 sinusoids, f1 = f2, A1=A2, traveling in the same direction, but different phase

1 0 2 0

1 2

y y sin kx t ; y y sin(kx t )

use trig identity: sin( ) + sin( ) = 2sin cos2 2

y y 2sin(kx t ) cos2 2

Constructive interference: = 0, ,…Destructive interference: = /2,3/2…

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10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 16

Case 2: 2 pulses traveling in opposite directions:

1801

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 17

1 12 2

Use trig identity again:

sin A sin B 2sin A B cos A B

ftxytxyftxytxy leftright λ

π2sin),(λ

π2sin),( 00

right left 02πxget y (x, t) y (x, t) 2y sin cos 2πft

λ

Standing wave:

Case 3. Standing waves. Two sinusoidal waves, same amplitude, same f, but opposite directions

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 18

2 22

2 2

standing

2standing 2

2

2 2standing

2

2 22

2 2

y ywave equation: v 0t x

2πxy (x, t) A sin cos 2πftλ

y 2πxCheck: 2πf Asin cos 2πftλt

y 2π 2πxCheck: A sin cos 2πftλ λx

y y 2π v A sint x

22 2x 2πcos 2πft 2πf v 0

λ λ

Equality satisfied iff fλ v. Okay? Yes.

Check that the standing wave satisfies the Wave Equation.

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10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 19

How to generate standing waves

1. Aim two loud speakers at one another. 2. Have two boundaries causing one wave to be

reflected back on itself. Each boundary is a node. The reflected wave travels in the opposite direction.

1808 Standing waves

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 1810 Nodes, in string

.....4,3,2,12

2sin:shapesspatialPossible

nLπnxA

n n

2πxStanding wave form: Asin cos 2πft

2L nv f n 1,2,3, 4...........n 2L

Standing waves between reflecting walls

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iclicker question after string demo:Which of the following statements are true about the string motions described above?

A. The human ear can directly hear the string vibrations.

B. The human ear could only hear the string vibration if it occurs in vacuum.

C. The human ear can only hear the string vibration if it produces a sound wave in air.

Demo:standing transverse waves in a string

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10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 22

Standing waves in air:

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Standing longitudinal waves in a gas-filled pipe:

Do Eric’s supersized fireplace log demo.

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Sound hole

frets

6 Strings (primary elastic elements)

Bridge(couples string vibration to sound board)

Sound board

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Coupling standing wave resonances in materials with sound:

iclicker question:Suppose an “A” is played on a guitar string. The standing wave on the string has a frequency (fg), wavelength (g) and speed (cg) . Which properties of the resultant sound wave are the same as wave on the guitar string?

A. Frequency fsB. Wavelength sC. Speed csD. A-CE. None of these

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Comments about waves in solids and fluids: Solids can support both transverse and

longitudinal wave motion. Earthquake signals

Fluids, especially gases, can support only longitudinal wave motion

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 271820 harmonics

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10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 28

The Doppler effect . Funny things happen if wave source and wave detector are moving with respect to one another.

Source.vs=0, freq f Observer, vO , f’ = ?

O' '

O

Let v' = speed of wave relative to observerv ' v v but wavelength is unchanged

so for source, v= f , but for observer, v = f f ' v '/

v vf '

' Ov vf f

v

1708 Doppler, motorcycle & car

Case 1. Source stationary, observer moving toward source.

Case 2. It’s easy to see that if observer moves away from source

' Ov vf f

v

What happens as observer goes past the source?

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Case 3. The Doppler effect for stationary observer, source moving toward observer A. Let v = velocity of sound, f = frequency of source

S S

' S

' '

'S S

During each vibration of source, source moves distance v T v / f.

vTherefore observed wavelength

fBut f v / for Observer A

v vso fv vvf f f

'

S

vf fv v

1709 Doppler, 2 observers, one source

'

S

vf fv v

Case 4. Souce moving away from observer B

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 30

S

OSO vv

vvff

:effectDoppler soundofSummary toward

away

R

RSO vv

vvff

: wavesneticelectromagfor effect Doppler Relative velocity of source toward observer

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10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 31

Example:

S

OSO vv

vvff

toward

awayS

OSO vv

vvff

:caseIn this

SOSO vvff Velocity of sound:

v = 343 m/s m/s40)( S

OSO ffvvvv

Case 4. Source and observer both moving.

o

oss

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 32

Doppler effect for electromagnetic waves For sound waves, model and equations used

velocity of source or observer with respect to the mediumFor electromagnetic (EM) waves (light, radio waves),

there is no medium and the velocity of light is c.The model and equations have to be different (Einstein).

' relative

relative

c vf f

c v

Applications: wavelengths of spectral lines from stars, e.g. H atom, are slightly shifted.

Doppler radar: make sketch on board

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iclicker question

Is Doppler radar described by the equations given above for sound Doppler?

(A)yes (B) no

Is “ultra sound” subject to the sound form of the Doppler effect?

(A) yes (B) no

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10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 34

10/31/2013 PHY 113 C Fall 2013 -- Lecture 18 35

right left 02πxy (x, t) y (x, t) 2y sin cos 2πft

λ

“Standing” wave: BABABA 2

12

1 cossin2sinsin

1810 Nodes & antinodes in string