PHYS 0212 Sound and Human
Hearing
1
Sound and Human Hearing
This experiment has five parts:
3. Study the relationship between frequency and musical scales
2. Determine the velocity of sound in air with an openopen tube
4. Measure the beat frequency between two sources that are close in pitch
1. Determine the velocity of sound in air with an openclosed tube
5. Measure the harmonics of your voice
The first two parts of this lab will use standing waves to determine the velocity of
sound.
First, let us consider standing waves on a string.
Waves on a string are
transverse
– the displacement is perpendicular to the
direction of travel.
Travel
y
x
Displacement
♫
PHYS 0212 Sound and Human
Hearing
2
y
x
(
)()
,s
i
n
yxt
A
t k
x
ω
=+
2
2
f
T
π
ωπ
==
2
k
λ
=
The equation for a wave is:
is the angular frequency:
k
is the wave number:
is the wavelength
This gives you the displacement
y
at a position
x
and at a time
t
.
The frequency
f
is the number of
crests that pass a given point per
second.
The units are Hertz (Hz)
where 1 Hz = 1/s.
The period T is the time for one
complete oscillation.
PHYS 0212 Sound and Human
Hearing
3
When a wave encounters a barrier, it reflects and travels back the way it came.
If
the boundary is fixed, the reflected wave will be inverted.
y
x
The incident and reflected waves will overlap and their displacements will add
together.
()
,
Incident wave
Reflected wave
We can use the trigonometric identity:
(
)
sin
sin
cos
cos
sin
A
BA
B
A
B
±=
±
To rewrite this as:
() ()
,2
s
i
n c
o
s
A
k
x
t
=
PHYS 0212 Sound and Human
Hearing
4
(
)
(
i
n
s
i
n
A
x A
x
ωω
−−
(
)
(
)
(
)
,
sin
cos
cos
sin
sin
cos
cos
sin
y x t
A
t
kx
t
kx
A
t
kx
t
kx
(
)
(
)
,
sin
cos
sin
cos
cos
sin
cos
sin
y x t
A
t
kx
t
kx
t
kx
t
kx
ωω ωω
=−
+
+
(
)
(
)
(
)
c
o
s s
i
n
A
t
k
x
=
Note that
y = 0
, when
kx
is a multiple of
π
:
,
0,1,2,
kx
n
n
…
If the boundary is placed at
x = L
, so that
L
is the length of the string, then
22
nn
Ln
k
πλ
=
This says that if
L
is equal to an integer multiple of a halfwavelength, both ends of
the string (
x = 0
and
x = L
) will always have zero displacement.
(
)
0,
0
(,) 0
yt
yLt
=
=
These two points are called
nodes
and the resulting
wave is called a
standing wave
.
1,2,3,
kL
n
n
=
=
…
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Hearing
5
X = 0
X = L
Node
Node
Antinode
n =
1
2
L
λ
=
The nodes are half a wavelength apart.
The antinode is the
x
position where the maximum amplitude occurs.
It is half
way between two nodes, so it is one quarter wavelength from either node.
2
4
PHYS 0212 Sound and Human
Hearing
6
vf
=
2
L
n
=
v
f
=
Relationship between wave velocity, frequency and wavelength.
n =
2
2
2
L
=
Antinode
Node
Node
Antinode
2
2
Node
2
L
n
=
22
vv
v
fn
L
nL
==
=
2
n
v
L
=
PHYS 0212 Sound and Human
Hearing
7
2
n
v
L
=
These frequencies are called
harmonics.
Note that if
n = 1
, then
1
2
v
f
L
=
Now we can rewrite the harmonics as
1
n
f
nf
=
This frequency is called the fundamental.
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 Spring '08
 Naples
 Physics, Frequency, Wavelength, Hz, kx

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