Geography Reference
In-Depth Information
PROBLEMS
Re
C exp (imx)
can be written
7.1.
Show that the Fourier component F (x)
=
as
F (x)
= |
C
|
cos m (x
+
x
0
)
|
and C
i
stands for the imaginary part of C.
7.2.
In the study of atmospheric wave motions, it is often necessary to consider
the possibility of amplifying or decaying waves. In such a case we might
assume that a solution has the form
m
−
1
sin
−
1
C
i
|
where x
0
=
C
ψ
=
A cos (kx
−
νt
−
kx
0
) exp (αt)
where A is the initial amplitude, α the amplification factor, and x
0
the initial
phase. Show that this expression can be written more concisely as
Re
Be
ik(x
−
ct)
ψ
=
where both B and c are complex constants. Determine the real and imaginary
parts of B and c in terms of A, α, k, ν, and x
0
.
7.3.
Several of the wave types discussed in this chapter are governed by equations
that are generalizations of the wave equation
c
2
∂
2
ψ
∂x
2
This equation can be shown to have solutions corresponding to waves of
arbitrary profile moving at the speed c in both positive and negative x direc-
tions. We consider an arbitrary initial profile of the field ψ ; ψ
∂
2
ψ
∂t
2
=
=
f(x)
at t
0. If the profile is translated in the positive x direction at speed c
without change of shape, then ψ
=
f(x
), where x
=
is a coordinate mov-
x
+
=
ing at speed c so that x
ct. Thus, in terms of the fixed coordinate
=
−
x we can write ψ
ct), corresponding to a profile that moves
in the positive x direction at speed c without change of shape. Verify that
ψ
f(x
=
f(x
−
ct) is a solution for any arbitrary continuous profile f(x
−
ct).
x
and differentiate f using the chain rule.
Hint
: Let x
−
ct
=
7.4.
Assuming that the pressure perturbation for a one-dimensional acoustic wave
is given by (7.15), find the corresponding solutions of the zonal wind and
density perturbations. Express the amplitude and phase for u
and ρ
in terms
of the amplitude and phase of p
.
7.5.
Show that for isothermal motion (DT /Dt
=
0) the acoustic wave speed is
given by (gH )
1/2
where H
=
RT /g is the scale height.
