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Sound Waves
Michael Fowler 3/23/07
“OneDimensional” Sound Waves
We’ll begin by considering sound traveling down a hollow pipe, to avoid unnecessary
mathematical complications. Sound is a longitudinal wave—as the wave passes through, the air
moves backwards and forwards in the pipe, this oscillatory movement is in the same direction the
wave is traveling.
To visualize what’s happening, imagine mentally dividing the air in the pipe, which is at rest if
there is no sound, into a stack of thin slices.
Think about one of these slices. In equilibrium, it
feels equal and opposite pressure from the gas on its two sides.
(This is analogous to the little bit
of string at rest feeling equal and opposite tension on its two sides, but of course the gas pressure
is inward).
As the sound wave goes through, the pressure wave generates slight differences in
pressure on the two sides of our thin slice of air, and this imbalance of forces causes the slice to
accelerate.
To analyze this quantitatively—to apply
Fm
a
=
G
G
to the thin slice of air—we must begin by
defining
displacement
, the quantity corresponding to the string’s transverse movement
( )
,
yxt
.
We shall use
( )
,
sxt
to denote the
horizontal
(along the pipe) displacement of the thin slice of air
which rests at position
x
when no sound is present.
.
An
animated
version of this diagram is available
here
!
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If the pipe has radius
a
, and hence crosssectional area
2
a
π
, a slice of air of thickness
x
Δ
has
volume
, so writing the density of air
2
ax
Δ
ρ
(1.29 kg/m
3
), the mass of the slice of air is
.
Clearly, its acceleration is
2
mV
a
ρρ
==
Δ
x
( )
2
,/
as
x
t
2
t
=
∂∂
, so we already have the right
hand side of
.
To find the left hand side—the force on the thin slice of air—we must
find the pressure imbalance between the two sides.
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 Fall '07
 MichaelFowler

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