PHYSICS 231

INTRODUCTORY PHYSICS I

Lecture 23

• Speed of sound in fluid

(for solid, replace )

• Intensity

• Intensity Level - dB

• Spherical Waves

Last Lecture -Sound

v =B

!

! = 10 log10I

Io

I = I010! /10

I =P

4!r2

!

B"Y

!

I =P

A

!

I0

=10"12W/m

2

Doppler Effect, Moving Observer

Fig 14.8, p. 435

Slide 12

Towards source:

Away from source:

ƒ' = ƒv + v

o

v

!"#

$%&

ƒ' = ƒv ! v

o

v

"#$

%&'

Fig 14.9, p. 436

Slide 13

v = speed of sound, vO = speed of observer

Doppler Effect:Source in Motion

! ' = ! " vsT

= ! " vs!

v

= ! 1" vs v( )

f ' = v! '

f ' = fv

v ! vs

!

" #

!

"

Doppler Effect, Source in Motion

Approaching source:

Source leaving:

f ' = fv

v ! vs

f ' = fv

v + vs

Example 14.6

An train has a brass band playing a song on a flatcar. Asthe train approaches the station at 21.4 m/s, a person onthe platform hears a trumpet play a note at 3520 Hz.DATA: vsound = 343 m/s

a) What is the true frequency of the trumpet?

b) What is the wavelength of the sound?

c) If the trumpet plays the same note after passing theplatform, what frequency would the person on theplatform hear?

a) 3300 Hz

b) 9.74 cm

c) 3106 Hz

Fig 14.11, p. 439

Slide 15

Shock Waves (Sonic Booms)

When the source velocity exceeds the speed of sound,

Application: speed radar

Application: weather radar

Both humidity (reflected intensity) and speed of clouds(doppler effect) are measured.

Doppler Effect:Both Observer and Source Moving

Switch appropriate signs if observeror source moves away

ƒ' = ƒv ± vo

v ± vs

!

"#$

%&

Example 14.7

At rest, a car’s horn sounds the note A (440 Hz). Thehorn is sounded while the car moves down thestreet. A bicyclist moving in the same direction at10 m/s hears a frequency of 415 Hz.DATA: vsound = 343 m/s.

What is the speed of the car? (Assume the cyclist isbehind the car)

31.3 m/s

Example 14.8a

A train has a whistle with a frequency of a 1000 Hz,as measured when both the train and observer arestationary. For a train moving in the positive xdirection, which observer hears the highest frequencywhen the train is at position x=0?

Observer A has velocity VA>0 and has position XA>0.Observer B has velocity VB>0 and has position XB<0.Observer C has velocity VC<0 and has position XC>0.Observer D has velocity VD<0 and has position XD<0.

Example 14.8b

A train has a whistle with a frequency of a 1000 Hz, asmeasured when both the train and observer arestationary. A train is moving in the positive xdirection. When the train is at position x=0,

An observer with V>0 and position X>0 hears afrequency:

a) > 1000 Hzb) < 1000 Hzc) Can not be determined

Example 14.8c

A train has a whistle with a frequency of a 1000 Hz,as measured when both the train and observer arestationary. A train is moving in the positive xdirection. When the train is at position x=0,

An observer with V>0 and position X<0 hears afrequency:

a) > 1000 Hzb) < 1000 Hzc) Can not be determined

Example 14.8d

A train has a whistle with a frequency of a 1000 Hz,as measured when both the train and observer arestationary. A train is moving in the positive xdirection. When the train is at position x=0,

An observer with V<0 and position X<0 hears afrequency:

a) > 1000 Hzb) < 1000 Hzc) Can not be determined

Standing Waves

Consider a wave and its reflection:

yright = Asin 2!x

"# ft

$%&

'()

*

+,

-

./

= A sin 2!x

"

$%&

'()cos2! ft # cos 2!

x

"

$%&

'()sin2! ft

012

345

yleft = Asin 2!x

"+ ft

$%&

'()

*

+,

-

./

= A sin 2!x

"

$%&

'()cos2! ft + cos 2!

x

"

$%&

'()sin2! ft

012

345

yright + yleft = 2Asin 2!x

"

$%&

'()cos2! ft

Standing Waves

•Factorizes into x-piece and t-piece •Always ZERO at x=0 or x=m!/2

yright + yleft = 2Asin 2!x

"

#$%

&'(cos2! ft

Resonances

Fig 14.16, p. 442

Slide 18

Integral number of halfwavelengths in length L

n!

2= L

Nodes and anti-nodes

• A node is a minimum in the pattern

• An antinode is a maximum

Fundamental, 2nd, 3rd... Harmonics

Fig 14.18, p. 443

Slide 25

Fundamental (n=1)

2nd harmonic

3rd harmonic

n!

2= L

Example 14.9

A cello string vibrates in its fundamental mode with afrequency of 220 vibrations/s. The vibrating segment is70.0 cm long and has a mass of 1.20 g.

a) Find the tension in the string

b) Determine the frequency of the string when itvibrates in three segments.

a) 163 N

b) 660 Hz

Beats

Interference from two waves with slightly differentfrequency

Beat Frequency Derivation

After time Tbeat, two sounds will differ by onecomplete cycle.

n1! n

2= 1

f1Tbeat ! f

2Tbeat = 1

Tbeat =1

f1! f

2

fbeat =1

Tbeatfbeat = f

1! f

2

Beats Demo

Standing waves in Pipes - Open both ends

Same expression for closed at both ends

!n= n

!

2

!n= (2n +1)

!

4

Standing waves in Pipes - Closed one end

Example 14.10

An organ pipe of length 1.5 m is open at one end.What are the lowest two harmonic frequencies?

DATA: Speed of sound = 343 m/s

57.2 Hz, 171.5 Hz

Example 14.11

An organ pipe (open at one end and closed at the other)is designed to have a fundamental frequency of 440 Hz.Assuming the speed of sound is 343 m/s,

a) What is the length of the pipe?

b) What is the frequency of the next harmonic?a) 19.5 cm

b) 1320 Hz

Interference of Sound Waves

Assume sources “a” and “b” are “coherent”. Ifobserver is located ra and rb from the two sources,

ra! r

b= n" formaximum

ra! r

b= (n +1 2)" forminimumra

rb

Source a Source b

Observer

Example 14.12

A pair of speakers separated by 1.75 m are driven by thesame oscillator at a frequency of 686 Hz. An observerstarts at one of the speakers and walks on a path that isperpendicular to the separation of the two speakers.(Assume vsound = 343 m/s)

a) What is the position of the last intensity maximum?

b) What is the position of the last intensity minimum?

c) What is the position of the first intensity maximum?

a) 2.81 m

b) 6.00 m

c) 27 cm