38 AIR MASS ANALYSIS 
tory of a cyclone (when the warm 
sector occludes) have not yet been 
attacked theoretically owing to the 
extreme difficulty of the mathemati- 
cal treatment. But the nascent wave- 
form has already been dealt with 
very thoroughly from a mathematical 
viewpoint. While it must be admitted 
that much still remains to be done 
on the mathematical analysis of the 
young cyclone, as will become ap- 
parent later, it has been proved, 
mainly by H. Solberg’), that waves 
postulated in the Norwegian (wave) 
theory of cyclone formation can and 
must exist in our atmosphere. 
In the following the results of the 
mathematical theory will be repre- 
sented in a descriptive form. Readers 
who desire to study the complete 
theory are referred to “Physikalische 
Hydrodynamik” by V. Bjerknes and 
his collaborators’). Here the various 
types of waves will be described 
which originate in a liquid or a gas 
due to different causes, and it will 
be shown how these different causes 
act together in the atmosphere to 
give origin to the cyclonic waves. In 
this way we follow somewhat the 
same line of attack as used by the 
theoretical meteorologists. Owing to 
the complexity of the problem it has 
been necessary first to disregard some 
of the influences acting upon the 
‘waves in order to study somewhat 
simpler conditions. 
GRAVITY WAVES 
A well known type are the waves 
at a water surface, or, to be more 
accurate, the waves at the boundary 
between water and air. For wave 
motions of this type, the gravity of 
the earth is the controlling force 
2Bjerknes, V., Bjerknes, J., Solberg, H., and 
Bergeron, T.: Physikalische Hydrodynamik 
Berlin, 1938, pp. 565-621; Hydrodynamique 
Physique. Paris, 1934, 3 vols., Chapt. 14, 
(vol. 2). 
(disregarding the small ripples which 
are subjected to capillary forces). 
To understand better the physical 
process in a gravitational wave con- 
sider a mass of water which is lifted 
upwards from the original position 
of equilibrium by a _ disturbance. 
This quantity of water has thus ac- 
quired a certain potential energy. 
While falling back to its original 
place its potential energy is trans- 
formed into kinetic energy of motion. 
The velocity increases until the water 
passes through the equilibrium posi- 
tion. However, it does not come to 
rest but owing to the acquired veloci- 
ty continues to move downwards 
until the kinetic energy is trans- 
formed again into potential energy. 
Then the mass element reverses its 
direction of motion and ascends. In 
this way a wave motion is set up 
which is damped out gradually only 
by friction. This description applies 
to standing waves since it takes only 
vertical motion into account. For 
progressive waves it would have to 
be modified slightly. Moreover, it 
is not quite satisfactory to single out 
one mass element and neglect the 
motion of the surrounding water 
masses. These defects not withstand- 
ing, it shows that there is always a 
transformation from kinetic to po- 
tential energy and back in a gravita- 
tional wave, analogous to the oscilla- 
tions of a pendulum. 
Similarly, the internal waves at 
the boundary between two fluid layers 
of different density are gravitational 
waves. Waves of the gravitational 
type are stable as long as the original 
density distribution is stable. Un- 
stable stratification is rarely found 
in the earth’s atmosphere except near 
the ground, for obvious reasons. 
Therefore, gravity tends to generate 
only stable waves. 
