422 
characteristics of the system of equations (1) and (2) are 
defined as lines along which the partial derivatives 
du/dt, du/dx, dw/dt, and dw/dx cannot be determined 
directly from the equations. These lines are very useful. 
Tt can be shown that these are lines along which dis- 
continuities in the partial derivatives occur. For in- 
stance, in the rapid steady-state flow of water along a 
curb, a small notch in the curb causes small discontinui- 
ties in the derivative of the height of the water and a 
line of disturbed water slants out and downstream from 
the notch. Such lines and similar ones in steady-state 
gas flows are called Mach lines. Thus, in a steady-state 
flow, disturbances along the characteristics can actually 
be seen. 
In order to find the characteristics of equations (1) 
and (2) a well-known equation from elementary calculus 
must be used. If w and w are functions of x and ¢, then 
du = SF dt + (3) 
and 
ow ow 
dw = aap OH ap a lh (4) 
Equations (1)—(4) are now looked upon as a system of 
linear algebraic equations for the unknown functions 
du/dt, du/dx, dw/dt, and dw/dx. They have the matrix 
Ai By Cy D, Ey 
Ao Bo C2 Do Ep 
dt dx 0 O du 
0 O dt dx dw 
From this it is seen that the value of du/dt at any point 
(x, t) is expressed in terms of w, w, x, and t by the 
equation 
(5) 
iy By Cy Dp Ay JB Ch Dy 
du _ |#a Ba Ce Ds uh dy By © Da) G 
ot du dx 0 0 ‘iE dp © © 
Give teas O © Ge ak 
The condition that du/dt be indeterminate is that the 
denominator and numerator of the right-hand side of 
equation (6) be zero. This can be expressed for the 
denominator as follows: 
(A.C, — A.C) da? 
— (A,D, — A2D; + BC, — B2C,) dadi (7) 
+ (B,D> a B2D;) dt? = 0 
This equation is solved (by using the quadratic for- 
mula) for dx/dt or dt/dx, whichever is desired. This is 
best done in specific cases where usually many of the 
coefficients, A,, By, etc., are zero, but the symbolic 
solutions will be useful: 
C, : = = (BdiL 5 
4 (8) 
AY 
Gz: dt = @loo6 
DYNAMICS OF THE ATMOSPHERE 
These equations define the two characteristics (C, , C_) 
of equations (1) and (2). Note that the characteristics 
are functions of u, w, x, and ¢ and not of their deriva- 
tives. 
The Equations of Compatibility. The numerator of 
equation (6) must be zero where the denominator is 
zero. This condition is expressed by writing out the 
determinant in the numerator to obtain 
(« : ae ‘dt AC ~ ps) 
-») 
- (03 saod 
D,C2) = 0. 
_ dwdx 
dt dt (Cs 
Since this is to be true along the characteristics, the ap- 
propriate values of dx/dt are substituted in the equation. 
Then two equations of compatibility, one for each of 
the two roots dx/dt of equation (7), are obtained, 
(at — Bray \(Cae =F D2) 
dt 
du 
— (3% = Fray \(Cray aa D;) (9) 
= 
aye 7 a+ (CDi — DiC) = 0 
dx 
along a oP and 
(nm _ i, a) (Cee) 
a (2% = Bi x) Cra. = Dy) (10) 
ar = a—(C,D; = C2D,) = 0 
dx 
along no 
Numerical Integration by the Method of Charac- 
teristics. The conditions (9) and (10) form the basis 
for the method of numerical integration which is called 
the method of characteristics. The approximation is 
made that if two points (¢;, x;) and (¢;, x;) are very 
close together on a curve, then the slope of the curve is 
da _ %j ~ % 
dt lig = fig 
Similar approximations are made concerning du/dt and 
dw/dt. The values of wu and w are given at two points 
(t,, v1) and (&, x.) not connected by a characteristic 
and it is required to find w and w at some other point 
(ts, 23) (not given) where it is unknown. We define 
ts and #3 as the intersection of the characteristic line 
with slope a, through (4, 2) and the characteristic 
line with slope a through (t:, x2). Therefore, ts and 
