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