Geography Reference
In-Depth Information
we obtain
L
2
L
2πsx
L
A
s
=
f (x) sin
dx
0
In a similar fashion, multiplying both sides in (7.2) by cos(2π nx/L) and inte-
grating gives
L
2
L
f (x) cos
2πsx
L
B
s
=
dx
0
A
s
and B
s
are called the
Fourier coefficients
, and
f
s
(x)
=
A
s
sin k
s
x
+
B
s
cos k
s
x
(7.3)
is called the sth
Fourier component
or sth harmonic of the function f(x). If the
Fourier coefficients are computed for, say, the longitudinal dependence of the
(observed) geopotential perturbation, the largest amplitude Fourier components
will be those for which s is close to the observed number of troughs or ridges
around a latitude circle. When only qualitative information is desired, it is usually
sufficient to limit the analysis to a single typical Fourier component and assume
that the behavior of the actual field will be similar to that of the component. The
expression for a Fourier component may be written more compactly by using
complex exponential notation. According to the Euler formula
exp (iφ)
=
cos φ
+
i sin φ
1)
1/2
where i
≡
(
−
is the imaginary unit. Thus, we can write
f
s
(x)
=
Re
[
C
s
exp (ik
s
x)
]
(7.4)
=
Re
[
C
s
cos k
s
x
+
iC
s
sin k
s
x
]
where Re[ ] denotes “real part of” and C
s
is a complex coefficient. Comparing (7.3)
and (7.4) we see that the two representations of f
s
(x) are identical, provided that
B
s
=
[
C
s
]
A
s
=−
[
C
s
]
Re
and
Im
where Im[ ] stands for “imaginary part of.” This exponential notation will generally
be used for applications of the perturbation theory below and also in Chapter 8.
7.2.2
Dispersion and Group Velocity
A fundamental property of linear oscillators is that the frequency of oscillation ν
depends only on the physical characteristics of the oscillator, not on the motion
itself. For propagating waves, however, ν generally depends on the wave number
of the perturbation as well as the physical properties of the medium. Thus, because
