THE PERTURBATION EQUATIONS IN METEOROLOGY 
are denoted by C and C* it follows that 
z= [Ce~° + C*e “| exp {zala — (c — U)t]}. 
Since, according to (58), 2 must vanish at the rigid 
lower boundary (c = 0), it follows that 
C= —C = & say. 
Consequently, 
x = [2K cosh ac] exp {iala — (¢ — U)t]}, 
z = [K sinh ac] exp {iala — (« — U)t]}, 
p = —QK{g sinh az — a(o — U)’ cosh ag] 02) 
X exp {zala — (¢ — U)t}}. 
In the upper layer the pressure is given by 
pi = —Q'{g(C’e™ + Ce *°) 
— alo — U')'(Cle* — C*e **)} 
x exp {iala — (« — U’)tl}. 
This expression must vanish at the free upper surface 
(c = #) according to (59), so that 
[g — a(o — U')'IC’e** 
K! 
= —|g--a(o = U') Ce FZ = Slr — a (« — U’)4], 
where K’ is a new constant introduced for convenience. 
Consequently, 
xz’ = iK'|g cosh a(¢ — H) 
+ a(o — U’)’ sinh a(e — H)] 
X exp {zala — (« — U’)t}}, 
' = K’[g sinh a(c — A) 
— a(« — U’)’ cosh a(e — H)| 
X exp {iala — (co — Ui}, 
' = —Q/K"[g — a (o — U’)4| sinh a(c — H) 
X exp {zala — (¢ — U’)t]}. 
At the internal boundary the two conditions (58) and 
(59) become 
pant i? 
ca for X =X’ and Z=Z' =h, 
Strictly speaking, these two conditions as well as the 
condition at the free upper surface are to be satisfied 
at the disturbed position of the boundary, but only 
an error of higher order is incurred as explained pre- 
viously (p. 408) if the condition is made to hold at the 
undisturbed position. The two conditions at the internal 
boundary give two homogeneous linear equations for 
K and K’, and again the determinant must vanish if 
nontrivial solutions are to exist. Thus the following 
equation is obtained: 
Qlg — a(« — U)’ coth ah][g — a(e — U’)’ coth ah’ 
= Q'Ig — ac — U’)4. (94) 
x 
| 
(93) 
3 
ll 
413 
An equation of this type is referred to as a secular or 
frequency equation. It permits the determination of 
the wave velocity, and thus the frequency, as a func- 
tion of the wave length 27/a and of the physical 
parameters of the system. It is hardly necessary to 
discuss this equation in any detail because the wave 
motion in a fluid system such as is being considered 
here is too well known. In order to illustrate stability 
investigations one special example may be considered, 
however, namely that both layers are so deep com, 
pared to the wave length that 
coth ah’ = coth ah = 1. 
In this case we may write 
Oe = ae = UNilg = oe = UF 
= Q'Ig + a(o— U')'Ilg — a(o — U’)’). 
A first solution of this equation is given by the follow- 
ing expression, 
1/2 
¢ =U + (°) 5 
a 
This value of o gives the wave velocity of surface waves 
in deep water. It is physically plausible that such waves 
can exist in an infinitely deep layer with a free upper 
surface, even if the lower boundary is an internal dis- 
continuity rather than a rigid plane, because the ampli- 
tude of the surface waves decreases to zero at the 
boundary. If this type of wave is disregarded, we ob- 
tain from (95) a quadratic equation for o whose roots 
are 
_UQ+ UY 
Q+ Q' 
(2 Q= aU ae 
aQ + Q’ QQ + Q’)? : 
This familiar expression shows that the wave velocity 
consists of two terms, the ‘‘convective”’ term which 
represents a weighted mean of the velocities in both 
layers and the ‘‘dynamic” term which depends on the 
density and the wind discontinuity. The latter term 
can evidently become imaginary for sufficiently small 
density differences and wave lengths or for sufficiently 
large wind discontinuities. Then we may write 
(95) 
(96) 
o 
(97) 
o = oi + 102, 
and the periodicity terms of the perturbation equa- 
tions become 
exp {zala — (o, — U)t] F ast}. 
Thus the perturbation quantities depend exponentially 
on the time and, since one of the two exponentials has 
a positive exponent, a solution exists which increases 
exponentially with time, indicating that the motion 
is unstable. The particular type of instability arising 
here is referred to as shearing instability since it is due 
to the wind shear. 
Once the solutions (90) with the various relations 
