Sound - University of California, Berkeleymoller.physics.berkeley.edu/~phys8a/lec34_sound.pdf04/16/2007 YGK, Physics 8A Sound •Most of the wave phenomena can be demonstrated with

Sound

Prof. Yury KolomenskyApr 16/18, 2007

04/16/2007 YGK, Physics 8A

What We Know About Waves• So far, we’ve learned how to

describe a single wave in a 1delastic medium Longitudinal and transverse waves Special case: sine waves

vk

!=

!

k =2"

#

!

" =2#

T


Superposition of Waves• Consequences:

Interference Constructive (add in phase) Destructive (add out of phase)

Beats Two sources with different

frequencies Lab this week !

Standing waves Superposition of waves

traveling in opposite directionsin finite medium

1 2( , ) ( , ) ( , ) y x t y x t y x t! = +


Standing Waves

• Add two waves propagating in oppositedirections Result is a wave that appears fixed in space

( ) [ ], 2 sin cosmy x t y kx t!" =


n n n

a

a

a

n

( ) [ ]

The displacement of a standing wave is given by the equation

, 2 sin cos

The position dependant amplitude is equal to

These are defined as positions

2 si

where the stand

:

ing

wa

n

ve a

m

my kx

y x t y kx t!" =

Nodes :

mplitude vanishes. They occur when 0,1, 2,

2 0,1, 2,...

These are defined as positions where the standing

wave amplitude is maximum.

1They occur when

2

2

n

kx n n

x n n nx

kx n

#

$

$$

#

= =

% = % =

& '= +()

=

Antinodes :

0,1, 2,...

2 1 0,1, 2,...

2

The distance between ajacent nodes and antinodes is /2

The

1

2 2

distance between a node and an ajacent antinode is /4

n

n

x n x nn

$

$$

#

#

#

#

=*+

& '% = +

& '" = +(% =( *) )+

*+

Note 1 :

Note 2 :


A

A

A

B

B

B

Resonances occur when the resulting standing wave

satisfies the boundary condition of the problem.

These are that the Amplitude must be zero at point A

and point B and arise from the fact that the strin

1

1

g is

clamped at both points and therefore cannot move.

The first resonance is shown in fig.a. The standing

wave has two nodes at points A and B. Thus 2

. The second standing wave is sh2 own

L

L

!

! =

=

"

2

in fig.b. It has three nodes (two of them at A and B)

In this case 2 2

L L! !!# $

= = "% &' (

=

3

The third standing wave is shown in fig.c. It has four nodes (two of them at A and B)

In this case 3 The general expression for the resonant2

2 wavelen

2

gths is: 1, 2,3,

3

..

n

LL

Ln

n

!

!

!!" #

= =

=

=$% &' (

= . the resonant frequencies 2

n

n

v vf n

L!= =


Boundary Conditions• (see blackboard)


Examples: Musical Instruments• Guitar• Organ• Bell


Sound• Most of the wave phenomena can be demonstrated

with sound• Sound waves: longitudinal waves in elastic materials

Can be solids or fluids, but we normally associate soundwith propagation through air


3d Picture• Radial propagation

Wavefront: locations of maxima Ray direction: normal to wavefront Mathematically:

!

"p(r,t) = A(r)sin(kr #$t)

A(r) =A0

r(for spherical waves)


Speed of Sound• The most basic parameter describing sound wave

propagation Property of the medium

Depends on elasticity and density of the material

!

v =B

"

6420Aluminum6000Granite1480Water970Helium (0oC, 1 atm)343Air (20oC, 1 atm)331Air (0oC, 1 atm)Speed of Sound (m/s)Medium

where B is bulk modulus and ρ is density


Consider a wave that is incident normally on a surface

of area . The wave transports energy. As a result

power (energy per unit time) passes through .

We define at the wav

A

P A

Intensity of a sound wave

2

e i

ntensi

ty the rat

SI units: W/m

io /

PI

A

I P A

=

22 2

The intensity of a harmonic wave with displacement amplitude is given by:

In terms of the pressure amplitude

Co

1

ns

ider a point source S emitting

. 2 2

a power in t

m

m m

s

P

vI s I p

v

! "

!

# $ # $= = %& ' & '

( )( )

he form of sound waves

of a particular frequency. The surrounding medium is isotropic so the waves

spread uniformly. The corresponding wavefronts are spheres that have S as

their center. The sound i

2

2ntensity at a distance from S is:

1The intensity of a sound wave for a point sources is proportional

t

4

o

r

r

PI

r*=

Intensity


The auditory sensation in humans is proportional to the logarithm of the

sound intensity . This allows the ear to percieve a wide range of

sound intensities. The threshold of hearing o

I

I

The decibel

12 2

0

is defined as the lowest

sound intensity that can be detected by the human ear. 10 W/m

The sound level is defined in such a way as to mimic the response

of the human 10loear. go

I

I

I

!

!

"

# $= % &

' (

=

( )/10

is expressed in decibels (dB)

We can invert the equation above and express in terms of as:

10

For we have: 10 log1 0

increases by 10 decibels every time increa

o

o

I

I I

I I

I

!

!

!

!

!

= )

= = =Note 1 :

Note2 :

4

ses by a factor of 10

For example 40 dB corresponds to 10o

I I! = =

The Decibel (dB)


Typical Sound Intensities

160Instant perforation of eardrum140Military jet takeoff130Typical threshold of pain110 (113 after Marleau’s goal)Sharks playoff game !100iPod/CD player at max level100Large orchestra80Vacuum cleaner70Busy street traffic60Normal conversation40Mosquito20Whisper10Rustling Leaves0Threshold of hearingIntensity (dB)Source


Sound Frequencies• Human ear is built for large dynamic range

rather than high precision Audible range: linear frequencies f ~20 Hz…20 kHz Below 20 Hz: infrasound (which dogs hate) Above 20 Hz: ultrasound (which bats and dolphins love)

• Musical note scale: logarithmic fn = f0*2n/12 where n is the step number relative to the

frequency f0 (base of the octave) This scale is logarithmic, because log2(fn)=log2(f0)+n/12 12 frets/octave on a guitar, 12 keys on a piano Example: f(C4)=256 Hz, f(C5)=512 Hz


vS

vS

vS

vS

vD

vD

vD

vD

D

S

v vf f f f

v v

+! != >

"

D

S

v vf f f f

v v

!" "= <

+

D

S

v vf f

v v

!" =

!

D

S

v vf f

v v

+! =

+

D

S

v vf f

v v

±! =

±

Doppler Effect

Sound - University of California, Berkeleymoller.physics.berkeley.edu/~phys8a/lec34_sound.pdf04/16/2007 YGK, Physics 8A Sound •Most of the wave phenomena can be demonstrated with

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