THE METHOD OF CHARACTERISTICS 
The values from equation (16), substituted in equations 
(9) and (10), give the results 
u + 2c = const along - =u+e (18) 
and 
dx 
u — 2c = const along ae ge (19) 
Thus the numerical or graphical computing scheme 
described for equations (1) and (2) can be performed 
using equations (18) and (19). c 
Simple Waves.! Since A, , etc., are not explicit func- 
tions of x and ¢ and HE, = EH, = 0, with proper initial 
conditions there can be simple waves in the model 
described by equations (14) and (15). This is a powerful 
method of investigating the results of certain initial 
conditions under an inversion. In addition, these equa- 
tions are a suitable set to make the idea of a simple wave 
clear. A simple wave results when w and h are constant 
along the straight characteristic lines 
dx 
dt 
Since these straight lines can be determined from the 
initial conditions, or conditions along any line in the 
x, t plane, wu and h can be found throughout a region of 
the x, ¢ plane by drawing the characteristic lines cover- 
ing that region. 
Expansion Waves. An expansion wave. under an in- 
version is best demonstrated by the following example. 
The air under an inversion is at rest in a channel formed 
by two mountain ranges which restrict motion normal 
to them. It is held at rest by meteorological conditions 
at the termination of the mountain ranges. If these 
conditions change so that the air under the inversion 
begins to flow out from between the ranges, an expan- 
sion wave results. Figure 3 shows the height of the inver- 
sion and the wind speed along a cross section through 
the center of the channel under these conditions. The 
flow described qualitatively in Fig. 3 can be described 
quantitatively in the a, ¢ plane (see Fig. 4). The obser- 
vation of w at A and the initial height of the inversion 
at rest form the boundary conditions that determine the 
flow to the right of A. These values of wu determine c 
and therefore w + c. Since w is constant along the lines 
dx/dt = u + c, we can compute it throughout the 
region between the mountains. This example empha- 
sizes the ease with which the flow can be discussed if it 
is made up of simple waves. Note from formula (19), 
which applies in this case, that under these conditions 
the inversion height decreases as the speed of flow out- 
ward from between the mountains increases, that is, an 
increasing speed of east wind is a decreasing wind in this 
coordinate system. An example of an expansion wave 
that has synoptic importance will be given in the dis- 
cussion of squall lines. It should be emphasized that 
=ut+e. (20) 
1. The word ‘“‘wave”’ is used here in its general sense, 
familiar to meteorologists in ‘‘cold wave,” ‘frontal wave,” 
etc. No periodicity or sinusoidal properties are implied. 
425 
the example of this paragraph is used for its definiteness 
and simplicity. Expansion waves in the atmosphere 
need not occur between mountain ranges and need not 
\SOGILZA 
Fig. 3.—Motion of an expansion wave shown in successive 
cross sections. 
be associated with air initially at rest. The whole sys- 
tem of flow (in particular, the zone of still air) can have 
a constant velocity of any direction superimposed on 
t CHARACTERISTIC 
: LINES 
> 
A x 
Fic. 4.—The characteristics of an expansion wave. 
it. The rigid boundary is not necessary. Any flow which 
is essentially one-dimensional can be studied by the 
means presented here. 
Compression Waves. If the flow at the mouth of the 
valley changes from zero velocity to a flow into the 
valley, a compression wave results. A cross section of 
such a compression wave for successive times is given 
in Fig. 5. The z, ¢ diagram for such a flow is given in 
Fig. 6. Again the values of wu and the inversion height at 
A are the boundary conditions that determine the flow. 
We have seen that a compression wave leads to an 
envelope of the characteristic lines. In this case the 
envelope cannot persist for any great length of time so 
that a pressure jump forms. 
