42 AIR MASS ANALYSIS 
frontal surface the pressure gradients 
and consequently the winds are like- 
wise different; these are observed 
facts. The observations also indicate 
strongly that the cyclones originate 
from unstable waves on such frontal 
surfaces. The possibility of unstable 
waves with the characteristic dimen- 
sions and motions found in nascent 
eyclones, however, has to be proved. 
The theory shows that very small 
waves are unstable up to a length of 
a few 100m, or, if the vertical strati- 
fication is not very stable, a few km. 
Exact figures can not, of course, be 
given as long as the density and 
wind distribution are unknown. The 
limit between. stable and unstable 
waves depends on the lapse rates of 
temperature in both air masses, and 
on the temperature and wind dis- 
continuities at the frontal surface. 
But it can be stated that for suffi- 
ciently small waves the shearing in- 
stability (wind discontinuity) is more 
effective than the gravitational sta- 
bility. In other words, very short 
waves are unstable and have more 
the character of shearing waves than 
of gravitational waves. For the theory 
of the cyclone these waves are evi- 
dently far too short. When the waves 
are longer the stabilizing influence 
of gravitation becomes more _ pro- 
nounced so that beyond a certain 
critical wave length the waves will 
be stable, because the effect of shear- 
ing, which tends to produce instability, 
decreases with increasing wave length 
and is over-compensated by the sta- 
bilizing effect of gravitation. The 
gravitational-wave character is now 
predominant. Billow clouds are waves 
of this type. 
The wave length of billow clouds 
is the greater, the smaller the tem- 
perature discontinuity. From this it 
has been concluded that the wave 
theory of cyclones breaks down since 
wave lengths of the order of 1000 
km could only exist for infinitely 
small temperature discontinuities and 
impossibly large wind discontinuities, 
according to the formula for billow 
clouds. The fallacy of this objection 
can now be seen immediately. In the 
investigation of the billow waves 
the influence of the earth’s rotation 
is neglected and can be neglected 
owing to the small dimensions. But 
waves of cyclonic dimensions are 
greatly modified by the rotation of 
the earth around its axis. 
Thus, the wave motion becomes 
with increasing wave length more and 
more inclined to the vertical under 
the influence of the deflecting force 
of the earth’s rotation. This leads 
to the formation of unstable waves 
in the following way. The stable 
character of gravitational waves in 
the earth’s atmosphere depends on 
the difference between the weight of 
the oscillating particle and that of 
its surroundings and thus on the ver- 
tical component of the oscillation. 
Consequently, as the waves become 
longer and the wave motion more 
horizontal the stabilizing effect de- 
creases, since the vertical component 
of motion decreases. The computa- 
tion shows that waves whose length 
is of the order of 1000 km are unstable. 
At such wave-lengths the shearing in- 
stability is greater than the gravita- 
tional stability (which is small) owing 
to the almost horizontal motion of 
the particles. But with still longer 
waves the shearing instability be- 
comes smaller than the dynamic sta- 
bility caused by the earth’s rotation 
and, therefore, such waves become 
stable again. 
It is obvious that the only waves 
which may be regarded as cyclonic 
are the unstable ones whose length 
is of the order of 1000 km. They 
are of a mixed type because, in addi- 
