414 
between the different constants have been found it is, 
of course, possible to satisfy any required initial con- 
ditions by suitable lmear combinations of expressions 
of the form (90). 
Wave Motions in a Compressible Atmosphere. As 
an example of motion in a compressible medium, the 
case of an atmosphere may be considered where the 
temperature decreases linearly with the elevation at 
the rate e. In the undisturbed state the atmosphere 
may be at rest. The lower boundary of this atmosphere 
may be at c = —h’; there may be an internal dis- 
continuity at c = 0 where the temperature, and con- 
sequently the density, changes abruptly. The upper 
surface is free. Its height is determined by the fact that 
temperature, pressure, and density all vanish here if 
the temperature decreases linearly with altitude. The 
earth’s rotation will again be neglected; this is per- 
missible for small-scale phenomena such as microbaro- 
metric oscillations or billow clouds. 
It follows from these assumptions that in the un- 
disturbed state the hydrostatic equation must be satis- 
fied: 
1 oP 
Q ac’ 
and that pressure, temperature, and density are func- 
tions of ¢ only, specifically, 
P T g/Re 
Po (7;) 
Q T (g/Re)—1 
Qo -(F) ; 
Further, we shall assume that changes of state are 
adiabatic. Then, according to (15a), the coefficient of 
piezotropy 
Pram ntes 
serial, 
Cn Re 
and, according to (10), the coefficient of barotropy 
_ 1— Re/g 
agers ey sire 
Both coefficients become equal if 
= 1 
— ee GIN\e aa = 
s=ca (ti, 
the adiabatic lapse rate. 
Since the motion may again be assumed as two- 
dimensional the perturbation equations are given ac- 
cording to (54) and (55), if the perturbation density 
is replaced by the perturbation pressure according to 
(56), 
(98) 
. (gz) = 
(@) + 9v(?) = 
sn 
7" ae Q 
Dias anil Oz 
Out 2 +2). 
DYNAMICS OF THE ATMOSPHERE 
The second of these equations may also be written in 
the form 
az 0 (p 0 Dp 
an BG (3) ag yO (5) con 
In general the system (99) has coefficients which de- 
pend on the elevation c. Only if the atmosphere is 
isothermal, « = 0, y and I are constants, and in this 
case the coefficients are constant if p/@ rather than p 
is considered as an unknown variable. The case of an 
isothermal atmosphere is therefore particularly easy 
to deal with. 
In the more general case of a nonisothermal atmos- 
phere the functional dependence of the unknown vari- 
ables x, z, and p on the height must be left open, and 
it may be assumed that 
x, 2, p = A(c), C(z), Dic) exp [ca(a — ot)]. (100) 
If these expressions are substituted in (99), the system 
may be transformed into a second-order ordinary dif- 
ferential equation for one of the three functions A, 
C, and D of c. For instance, the differential equation 
for the pressure amplitude D becomes 
dD g ae 
P+ (E-J\(a)e 
2 1 ao s Rg = 
+{-2 + ppl + @- 0] hao. 
When D is known, the vertical amplitude C can be 
obtained from the following expression, 
(101) 
o< ae 
Similarly, the horizontal amplitude A can be expressed 
by D but this function need not be known to satisfy the 
boundary conditions. 
The differential equation (101) can with the aid of 
the substitutions D = e“’X(c) and y = —2aT/e be 
transformed into the confluent hypergeometric equa- 
tion, 
ax lu-3 ie 
Y age Y RE tins dy 
ao 
- [2% + 
The case of one atmospheric layer has been discussed 
by Lamb [13] and V. Bjerknes and collaborators [4]. 
In the case of an internal discontinuity surface where 
such phenonema as billow clouds arise it may be as- 
sumed that the wave motion does not extend far up- 
ward or downward from the interface. An approximate 
solution may then be obtained in the following manner. 
Since 
; (103) 
— "| x = 0. 
2Qaec 
multiplication of (101) by 1 — «c/T> permits one to 
