Geography Reference
In-Depth Information
observer moving at the
phase speed
c
ν/k. This may be verified by observing
that if phase is to remain constant following the motion,
≡
Dφ
Dt
=
D
Dt
k
Dx
(kx
−
νt
−
α)
=
Dt
−
ν
=
0
ν/k for phase to be constant. For ν>0 and k>0
we have c>0. In that case if α
Thus, Dx/Dt
=
c
=
ct), so that x must increase
with increasing t for φ to remain constant. Phase then propagates in the positive
direction as illustrated for a sinusoidal wave in Fig. 7.2.
=
0, φ
=
k (x
−
7.2.1
Fourier Series
The representation of a perturbation as a simple sinusoidal wave might seem an
oversimplification since disturbances in the atmosphere are never purely sinu-
soidal. It can be shown, however, that any reasonably well-behaved function of
longitude can be represented in terms of a zonal mean plus a
Fourier
series of
sinusoidal components:
∞
f
(
x
)
=
(
A
s
sin k
s
x
+
B
s
cos k
s
x
)
(7.2)
s
=
1
2πs/L is the
zonal
wave number (units m
−
1
), L is the distance
around a latitude circle, and s, the
planetary
wave number, is an integer designating
the number of waves around a latitude circle. The coefficients A
s
are calculated
by multiplying both sides of (7.2) by sin(2π nx/L), where n is an integer, and
integrating around a latitude circle. Applying the orthogonality relationships
L
where k
s
=
0,
2πsx
L
2πnx
L
s
=
n
sin
sin
dx
=
=
L/2,s
n
0
Fig. 7.2
A sinusoidal wave traveling in the positive x direction at speed c. (Wave number is assumed
to be unity.)
