Geography Reference
In-Depth Information
As in Chapter 7 we assume that wave-like solutions exist of the form
Ae
ik(x
−
ct)
, ψ
T
=
Be
ik(x
−
ct)
ψ
m
=
(8.17)
Substituting these assumed solutions into (8.15) and (8.16) and dividing through
by the common exponential factor, we obtain a pair of simultaneous linear algebraic
equations for the coefficients A, B:
ik
c
β
A
U
m
k
2
ik
3
U
T
B
−
+
−
=
0
(8.18)
ik
c
U
m
k
2
2λ
2
β
B
ikU
T
k
2
2λ
2
A
−
+
+
−
−
=
0
(8.19)
Because this set is homogeneous, nontrivial solutions will exist only if the deter-
minant of the coefficients of A and B is zero. Thus the phase speed c must satisfy
the condition
=
U
m
) k
2
k
2
U
T
(c
−
+
β
−
U
T
k
2
2λ
2
U
m
)
k
2
2λ
2
+
0
−
−
(c
−
+
β
which gives a quadratic dispersion equation in c:
c
U
m
2
k
2
k
2
2λ
2
2
c
U
m
β
k
2
λ
2
−
+
+
−
+
+
(8.20)
β
2
U
T
k
2
2λ
2
k
2
+
−
=
0
which is analogous to the linear wave dispersion equations developed in Chapter 7.
The dispersion relationship in (8.20) yields for the phase speed
β
k
2
λ
2
+
δ
1/2
c
=
U
m
−
k
2
k
2
2λ
2
±
(8.21)
+
where
U
T
2λ
2
k
2
β
2
λ
4
k
4
k
2
−
k
2
2λ
2
δ
≡
2λ
2
2
−
+
+
We have now shown that (8.17) is a solution for the system (8.15) and (8.16) only
if the phase speed satisfies (8.21). Although (8.21) appears to be rather complicated,
it is immediately apparent that if δ < 0 the phase speed will have an imaginary part
and the perturbations will amplify exponentially. Before discussing the general
physical conditions required for exponential growth it is useful to consider two
special cases.
