412 DYNAMICS OF THE ATMOSPHERE 
specific meteorological problems. Even the method is 
discussed only to the extent to which it illustrates the 
application of the perturbation equations. The addi- 
tional procedures of mathematical physics which lead 
to a solution of the problem are mentioned only very 
briefly. 
Only a few fairly simple examples of the use of the 
perturbation method are given here which illustrate 
the various types of problems that can be handled in 
this manner. 
Perturbations in Two Incompressible Fluid Currents, 
in the Lagrangian System. As a first example of the 
application of perturbation equations to fluid motions, 
a very simple case will be considered. This case has 
been discussed in the textbooks on hydrodynamics, 
although as a rule only with the aid of the Eulerian 
system. Here the Lagrangian equations will be used. 
Assume that we have two incompressible homogeneous 
fluid layers, both in constant horizontal motion. Let 
the density of the lower fluid be Q, its velocity U, 
while the density and velocity for the upper fluid 
are Q’ and U’, respectively. The earth’s rotation will 
be neglected. The effects of capillary forces which are 
important for short water waves will also be disre- 
garded. Further, let the lower boundary be horizontal 
and rigid at Z = c = 0, let the internal discontinuity be 
at the level h, and the free surface be at H, while the 
thickness of the upper layer may be denoted by h’. 
The last two statements imply that both the internal 
and the free surface are horizontal in the undisturbed 
case. This is physically obvious and will be corrob- 
orated by the equations for the undisturbed motion. 
It is sufficient to consider the motion in a vertical 
XZ plane only. It is 
X=a+ Ut, Ge = @. (81) 
The equations of motion and of continuity reduce, 
according to (47) and (49) to 
d= =, (82) 
0=-55--4 (83) 
Q = Q = const. (84) 
The subscript zero for the density can, of course, be 
omitted. Equations analogous to (81)—(84) hold for 
the upper layer. Since, according to (82) and (83), 
the pressure is a function of the vertical coordinate c 
only, it follows from the boundary condition (53) that 
the internal and free surfaces must both be horizontal. 
If (83) is integrated and if the outside pressure at the 
free surface vanishes, 
P’ = gQ'(H — c), (85) 
P = gQ'h' + gQ(h — c). (86) 
The integration constants have been chosen so that 
the two equations satisfy also the condition that the 
pressure be continuous at the internal surface. Since 
according to (53) at the free surface P’ = 0, and at 
ll 
ll 
the internal surface P — P’ = 0, it follows that both 
these surfaces must be horizontal in the undisturbed 
case. 
The perturbation equations of motion and of con- 
tinuity follow from (54) and (55), namely, 
ax _l@p_ @ 
om Of@ aa (92); (87) 
Oi wleop uae 
a O80 Be (92); (88) 
Ox 0z 
and analogous equations follow for the upper layer. 
From this system of three equations the three unknown 
variables x, z, and p can be determined as functions 
of the coordinates at the time ¢t = ¢ , namely a and 
c, and of the time ¢. Since the system is not only linear, 
but also has constant coefficients, exponential or trig- 
onometric functions will represent solutions. It may 
be assumed that the functional dependence on a, c, 
and ¢ can be expressed in the form 
x, z,p = A, C, Dexp {ta(X — at) + BZ} 
= A,C, D exp {tala — (c — U)t] + Be}, Oo) 
where A, C, and D are constants. The complex form 
for the periodicity term is more convenient than the 
trigonometric form and is therefore used here. For 
the subsequent physical interpretation either the real 
or the imaginary part of the solution or a linear com- 
bination of both can be used. The form of the solution 
given above represents a wave in the a direction of 
the length 27/a@ and moving with the speed c. Special 
attention should be called to the appearance of U 
(and U’ for the upper layer) in the exponent since 
the perturbation quantities remain small only with 
regard to the undisturbed position of the particle, but 
not with respect to its znitzal position. 
The expressions (90) do not satisfy the condition 
that the perturbation quantities vanish at the time 
t = ft but it can be shown by transition to the cor- 
responding Eulerian equations that the expressions de- 
veloped here represent solutions of the problem. 
Substitution of (90) into equations (87)—(89) leads 
to a system of linear homogeneous equations for A, 
C, and D. The system has nontrivial solutions only if 
its determinant vanishes. It follows that B = +a. 
Two of the three constants A, C, and D may now be 
expressed by the third, and in view of the condition 
to be satisfied at the horizontal lower boundary, A and 
D may be expressed by C. Thus 
2 2 
A= seo! i (91) 
and analogous expressions hold for the upper layer. 
Because of the two values for 6 there are two different 
solutions, and a linear combination of the two solu- 
tions represents also a solution of the system of differ- 
ential equations. When the two arbitrary constants 
