Biomedical Engineering Reference
In-Depth Information
∂
2
p
∂t
2
=
c
2
∇
2
p,
(2.8)
which is the three-dimensional version of (1.7).
Solving this equation for given initial and boundary conditions is not
trivial. However, two highly symmetric cases have been thoroughly discussed:
the plane wave and the spherical wave. These two read
p
(
x
,t
)=
Ae
i
(
k
·
x
−wt
)
plane
,
(2.9)
p
(
x
,t
)=
A
r
e
i
(
kr−wt
)
spherical
,
(2.10)
where
r
is the modulus of the radius vector
x
. In these equations, we write
the pressure in terms of complex numbers. The physical quantities we are
interested in are the real parts of these expressions. The rationale behind this
trick is that taking time or spatial derivatives of harmonic functions in this
formalism is as simple as multiplying or dividing by a (complex) number. Let
us see this at work. In either geometry, it is possible to derive a relationship
between the particle velocity at any point and the pressure fluctuations there,
by means of (2.6) or (2.7). Since these equations are linear, we can be sure
that a solution with both
p
and
v
oscillating with the same frequency is
possible. However, a phase difference between them may appear that depends
on the symmetry of the solution, which in turn depends on the geometry
of the sources and boundaries. For example, in the planar case, where the
spatial derivative is everywhere equivalent to a multiplication by a complex
number
ik
, and the time derivative is equivalent to a multiplication by
iω
,
the pressure and the velocity are in phase because the coe
cient relating
them turns out to be real. However, this is not the case for a spherical wave.
Relating
p
and
v
by means of (2.7), we now obtain
ρ
0
iω
v
=
−
r
+
ik
p
e
r
,
1
−
(2.11)
which means that the pressure and velocity are no longer in phase. Mathe-
matically, this is reflected in the fact that the coe
cient relating them is a
complex number
α
=(
1
/r
+
ik
)
/
(
iωρ
0
).
Why are we saying that a complex coe
cient relating
p
and
v
represents
a phase difference between them? First, a real coe
cient means that
p
and
v
have zero phase difference, since the value of
p
at a given instant is that of
v
(up to a scale factor). On the other hand, the effect of a purely imaginary
coe
cient relating
p
and
v
can be seen as the fact that the value of
p
depends
on the value of the
time derivative of
v
(scaled by the imaginary coe
cient),
which is
π/
2 out of phase with respect to
v
. This is a consequence of the
following relationship, valid for complex harmonic functions:
−
iω
v
=
d
v
dt
.
(2.12)
