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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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
04/16/2007 YGK, Physics 8A
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! = +
04/16/2007 YGK, Physics 8A
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!" =
04/16/2007 YGK, Physics 8A
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 :
04/16/2007 YGK, Physics 8A
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
!
!
!!" #
= =
=
=$% &' (
= . the resonant frequencies 2
n
n
v vf n
L!= =
04/16/2007 YGK, Physics 8A
Boundary Conditions• (see blackboard)
04/16/2007 YGK, Physics 8A
Examples: Musical Instruments• Guitar• Organ• Bell
04/16/2007 YGK, Physics 8A
Sound• Most of the wave phenomena can be demonstrated
with sound• Sound waves: longitudinal waves in elastic materials
Can be solids or fluids, but we normally associate soundwith propagation through air
04/16/2007 YGK, Physics 8A
3d Picture• Radial propagation
Wavefront: locations of maxima Ray direction: normal to wavefront Mathematically:
!
"p(r,t) = A(r)sin(kr #$t)
A(r) =A0
r(for spherical waves)
04/16/2007 YGK, Physics 8A
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
04/16/2007 YGK, Physics 8A
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
04/16/2007 YGK, Physics 8A
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)
04/16/2007 YGK, Physics 8A
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
04/16/2007 YGK, Physics 8A
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
04/16/2007 YGK, Physics 8A
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
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