The Wave Equation
1
.
Acoustic Waves
We consider a general conservation statement for a region
U
R
3
containing a fluid which
is flowing through the domain U with velocity field
V
V x
,
t
. Let
x
,
t
denote the
(scalar) fluid density at
x
,
t
, and let
F
F x
,
t
denote the fluid flux at
x
,
t
.
Then
F x
,
t
x
,
t V x
,
t
describes the direction and speed of the fluid flow at
x
,
t
.
Proceeding
as we have in previous examples, we obtain the following equation asserting that the fluid
mass is conserved during the flow
t
x
,
t
div
x
,
t V x
,
t
0
for all
x
,
t
U
0,
T
This is another special case of the equation
t
u
divF
s
0
we have seen before, this
time with
u
,
F
V
,
and s
0.
This is one equation for four unknowns,
and the
3
components of V
.
An additional equation is obtained from the assertion that
momentum is conserved during the flow. This conservation statement, that the time rate of
change of momentum equals the sum of the applied forces, can be expressed in terms of
the state variables by the vector equation,
d
/
dt
B
x
,
t V x
,
t dx
B
pn dS x
,
where B denotes an arbitrary ball in U and
p
p x
,
t
denotes the scalar pressure field in the
fluid. Then by an integral identity that is related to the divergence theorem,
B
pn dS x
B
pdx
,
we arrive at
d
dt
x
,
t V x
,
t
t
x
,
t V x
,
t
V
x
,
t V x
,
t
p
.
This adds three equations to the system but also adds a new unknown, p, so the unknowns
now consist of
,
V
1
,
V
2
,
V
3
,
and p
.
To complete the system we add the so called equation of
state, a constitutive equation which asserts that
p
f
,
where f denotes a fluid dependent function relating pressure to density.
In one dimension, this system has the form
1
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t
x
,
t
x
x
,
t V x
,
t
0,
t
V
V
x
V
x
p
0,
p
f
(1.1)
This is a system of nonlinear first order equations. The solution of this system is, in general,
quite difficult, even in one dimension. Therefore we consider the simpler problem of
modeling the propagation of acoustic waves in the fluid. Acoustic waves are small
amplitude perturbations in the density field in a quiescent fluid. That is,
V x
,
t
0
x
,
t
where


1
x
,
t
0
1
x
,
t
where
0
const
and
 
1
p x
,
t
f
f
0
f
0
0
p
0
0
f
0
x
,
t
.
These equations express that the unperturbed velocity and density fields are equal to zero
and
0
const
,
respectively, while the perturbations in these fields,
and
,
are much less
than 1 in magnitude. The perturbation in the pressure field is determined from the density
perturbation and the equation of state.
Substituting these expressions into the equations
1.1
and neglecting any terms that
involve products of perturbations, leads to
t
x
,
t
x
x
,
t
0
t
x
,
t
f
0
x
x
,
t
0.
Then it is easy to show that both
x
,
t
and
x
,
t
satisfy the same second order equation,
tt
u x
,
t
a
2
xx
u x
,
t
0,
1.2
where
f
0
a
2
in this case. Equation (1.2) is referred to as the
wave equation
due to
the fact that its solutions exhibit wavelike behavior.
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 Spring '08
 Brewer
 Flux, Partial differential equation, wave equation, Yx, Ýr Þ, uÝx

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