14
I. Introduction
where
U}
- f
D'
0
n, CO n·
B= (
f
0
0
fV/g -tot«
U
A2 =
0 V
0
o H
As diseussed in eonneetion with (1.10), the preeeding system will be hyperbolie if any linear eombination of the coefficient matrices, kl AI + k2A2 , can be
transformed to a real diagonal matrix through multiplieation by bounded transformation matriees. Sueh transformation matriees always exist when the coefficient matriees are symmetrie . Thus an easy way to demonstrate that the preceding
system is hyperbolie is to exhibit a ehange of variables that renders AI and A2
symmetrie. A suitable transformation is obtained by letting v = 8- l u, where
c O
8- 1 =
0 c
( o 0
and c(x , y) = JgH . Then (1.28) beeomes
OV
- + AI -
- ov
+
- ov
A2- + Bv
- = 0,
01
ox
oy
where
The symmetry of AI and A2 imply that the linearized shallow-water equations are
a hyperbolie system.
The wave solutions to this hyperbolic system do not, however, propagate exaetly along the eharacteristic curves unless f is zero. The relationship between
the paths followed by propagating waves and the eharaeteristics is most easily
investigated by considering plane waves propagating parallel to the x -axis in a
basic state with no mean flow. Let the Coriolis parameter have the eonstant value
fo and define a veetor of new unknown funetions
v = ( u -
u + ghfc
) ,