Civil Engineering Reference
In-Depth Information
2.7
The complex exponential representation
In the complex representation, the function cos
(ωt
−
kx)
is replaced by exp[
j(ωt
−
kx)
].
The function exp[
−
j(ωt
−
kx)
] is removed from the cosine which is given by
exp[
j(ωt
−
kx)
]
+
exp[
−
j(ωt
−
kx)
]
cos
(ωt
−
kx)
=
(2.32)
2
and the remaining function is multiplied by two. The result, in signal processing, is called
an analytical signal (Papoulis 1984). It is simpler to handle this quantity rather than the
initial function, as the negative frequencies have been removed. Many authors use the
time dependence exp
(
jωt)
, where the positive frequencies are removed. The complex
representation of a wave travelling in the direction of positive
x
will be exp[
j(k
∗
x
−
−
ωt)
], because decreasing the amplitude in the direction of propagation implies that the
new wave number is the complex conjugate of
k
. When adding the positive frequency
components of a real signal to its negative frequency components, the initial real signal
must be obtained. This will be possible simultaneously for both pressure and velocity, the
characteristic impedances in both representations being complex conjugate. For instance,
the real damped wave characterized by
p(x,t)
=
A
exp
(x
Im
k)
cos
(ωt
−
x
Re
k)
(2.33)
υ(x,t)
=
A/
|
Z
c
|
exp
(x
Im
k)
cos
(ωt
−
x
Re
k
−
ArgZ
c
)
(2.34)
has the following two representations:
p
+
(x,t)
=
A
exp[
j(ωt
−
(
Re
k
+
j
Im
k)x)
]
(2.35)
υ
+
(x,t)
=
(A/Z
c
)
exp[
j(ωt
−
(
Re
k
+
j
Im
k)x)
]
(2.36)
p
−
(x,t)
=
A
exp[
j(
−
ωt
+
(
Re
k
−
j
Im
k)x)
]
(2.37)
(A/Z
c
)
exp[
j(
υ
−
(x,t)
=
−
ωt
+
(
Re
k
−
j
Im
k)x)
]
(2.38)
The quantities
p
−
and
p
+
are related by
(p
+
+
p
−
)/
2
=
A
exp
(x
Im
k)
cos
(ωt
−
x
Re
k)
(2.39)
In the same way
(υ
+
+
υ
−
)/
2
=
υ(x,t)
(2.40)
The characteristic impedance
Z
c
becomes
Z
c
for the time dependence exp
(
−
jωt)
.
The impedances present the same property.
Similar arguments about the reconstruction of a real signal can be used to demonstrate
that the bulk moduli
K
in both representations are related by complex conjugation; the
same is true for the density
ρ
.
References
Allard, J. F., Bourdier, R. and L'Esperance, A. (1987) Anisotropy effect in glass wool on normal
impedance at oblique incidence.
J. Sound Vib
.,
114
, 233 -8.
