54
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 wave speed. Solutions to the sinusoidally-forced
system have the form,
℘=℘
+℘
−+
A
itk
x
B
x
ee
()
ωω
,
(
2
)
which describes both forward (in +
x
direction) and backward moving waves with radian
frequency
, ω
, and wavenumber,
k
=
2
π
λ
.
(
3
)
Since
λ
is the wavelength,
k
can be interpreted as 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 effects 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.