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Exp#8: Acoustic Plane Waves
1. Properties of acoustic plane waves
Acoustic waves are low amplitude pressure waves that propagate through a fluid
according to the balance between inertia and compressibility.
They are governed by the
wave equation,
∂
2
2
2
2
2
℘
=
℘
t
c
x
,
(
1
)
where
℘
is the fluctuating pressure, and
c
is a wavespeed.
Solutions to the sinusoidally
forced system have the form,
℘=℘
+℘
−+
A
itk
x
B
x
ee
()
ωω
,
(
2
)
which describes both forward (in +ve
x
) and backward moving waves with radian
frequency
, ω
, and wavenumber,
k
=
2
π
λ
.
(
3
)
λ
is the wavelength, and so
k
can be seen to be the spatial equivalent of the radian
frequency,
ω
.
At any particular point in space and time, the wave phase,
φ
, is
ω
=
−
tk
x
,
(
4
)
and if
is fixed, then
kx
=
t
+
C
1
, where
C
1
is some constant phase offset.
For
convenience, let
= 0, then
C
1
= 0,
kx
=
t
, and
c
x
==
.
(
5
)
Similarly, from eq.(4) at fixed
t
(for example,
t
=0) then
=
kx
,
(
6
)
and so
∂φ
x
k
=
.
(
7
)
k
is the rate at which
φ
increases with
x
, and it is a constant.
Note that eq.(7) is
equivalent
to eq.(3), where differences
Δφ
and
x
Δ
are taken over one complete wave
cycle.
Acoustic
plane
waves are convenient theoretical objects, which vary and propagate in
only
one
direction.
It is possible to generate something like a plane wave by confining
the acoustic disturbances to a cylindrical tube.
If the wavelength,
λ
, is large compared to
the tube diameter, if all reflected waves (
i.e.
backward traveling components of eq.(2))
are small in amplitude compared to the forward traveling waves, and if boundary layer
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View Full Documenteffects and viscous losses are ignored, then
k
can be measured by eq.(7) for a system
forced at a known frequency,
ω
, when eq.(5) gives the speed of sound in air.
There is nothing in the wave equation that confines solutions to sines and cosines, and a
more general solution to eq.(1) than eq.(2) is
℘=
−
+
+
fc
t x
gc
(
)
(
)
.
(
8
)
Once again,
c
is the wave speed, and
f
and
g
are unknown functions describing waves
moving in positive, and negative
x
respectively.
2. Generating and measuring sound waves
While the fluid can, in principle, support wave motions with arbitrary shape (eq. 8),
generating arbitrary pressure fluctuations is not so simple.
The most common device for
making sound waves is the loudspeaker, where fluctuations in voltage and current across
a moving coil generate a force (through Maxwell’s equations) that moves the coil in one
direction.
The coil itself is attached to a cardboard cone of carefullyselected stiffness
and suspension.
Motion of the coil translates to motion of the cone, which imparts
motion to the air next to it.
Through the balance between inertia and compressibility,
vibrations from the speaker propagate away at the speed of sound.
All speakers represent
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 Fall '07
 Pottebaum

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