THE METHOD OF CHARACTERISTICS 
23 can be found by solving the two simultaneous equa- 
tions 
iy oe Pal 
i = ta 
203) mel ae 
= a+; pais 
These values of ¢; can be substituted im the equations 
(9) and (10): 
( pee Bray )(Coay — Dz) 
1 
a (2. Bos: a aaa Bea.) (Cray ii D;) 
i 
SN se (CHD DAO) (0) 
ip = th 
t3 — te 
U3 — Us 
| (i eee 
ig = ip 
Wz — We 
ae] Fog ee = D,C2) 
3 42 
(aes — Pra \(Csa | Da») 
— Bea) (Cra_ — Di) 
ll 
0, 
and these two equations can be solved for uz and ws 
because tf; and «3; are known. By proceeding in this 
manner from a line (such as the boundary of a flow, or 
an initial state) in the z, ¢ plane, along which w and w 
are known, values of wu and w over a portion of the 
(x, t) plane can be found. For instance, in Fig. 1 the 
t 
> 
x 
Fie. 1.—The region in which wu and w can be computed 
from values given on L. 
data between A and B on L allow a computation of the 
values of w and w in the hatched region R. For a full 
discussion of such “domains of dependence” and the 
related “regions of influence” the reader is referred to 
Courant and Friedrichs [5]. This method of integrating 
even the most complicated of such equations is easy 
to derive but is difficult to carry out in practice. Some 
simpler problems are easier to discuss. 
Simple Waves. If, in equations (1) and (2), H, = HE» 
= 0 and A,, A», By, Bz, ete., are not functions of 
x or t but only functions of w and w, then the factor 
1/dt can be removed from equations (9) and (10). 
These equations then define two functions of w and w 
(in implicit form, to be sure): 
Huw) = Ky 
Liuyw) = K_ 
along Ci, 
along C_, 
423 
where in general the constants K, are different for 
each C, characteristic and similarly for K_ and C_. 
Courant and Friedrichs have shown that a region of 
constant wand w, say Ww and wo, is separated from a 
region of varying w and w by a characteristic. Assume 
that this is a C, characteristic. Smee C_ characteristics 
intersect C characteristics, some of the C_ characteris- 
tics are common to regions in which uw and w are con- 
stant (w and wo) and to regions in which they vary. 
Along such C_ characteristics we can say 
L(u, w) = L(uo , wo) along C_. 
This is true along every C_ characteristic extending into 
the region of constant w and w. Thus along such C_ 
characteristics wis a function of w only. Since these C_ 
characteristics cover all of a certain region, we need 
only have the assurance that we are considering motion 
inside that region to know that wu is a function of w 
only. Inside such a region, if w remains constant along 
a line, wu remains constant and vice versa. 
We now focus our attention on the C, characteristics 
in this region where w is a function of w only. We can 
say w = k(w); we then have 
A(u, k(u)) = M(u) = Ky 
This shows that w is constant along C, . From equation 
(6) the slope of C, is given by 
dx = 
dt as 
along Cy. 
Now we have assumed (for simple waves) that a; is a 
function of uw and w. Since w has been shown to be 
constant along C, , and therefore w is constant along 
C,, this means that a; is constant along C, or that C,. 
has a constant slope and is then a straight lime. When a 
flow has a family of straight characteristics it is called 
a simple wave. Much of Courant and Friedrichs’s book 
Supersonic Flow and Shock Waves is concerned with 
simple waves and the reader is referred to it for a com- 
prehensive discussion of them. 
The two basic types of simple waves are the expansion 
wave and the compression wave. The names are derived 
from gas dynamics, but apply equally well to the expan- 
sion or compression of the distance between two 
straight characteristic lines in the general case pre- 
sented above. If the given values of one of the depend- 
ent variables wu and w, say u, along a line L are such 
that the straight characteristics in a simple wave diverge 
as they move away from JL, then there is an expansion 
wave in that region (see Fig. 4). Note that, smce w is 
constant along these lines, as we move away from L 
along a characteristic, the distance to the nearest point 
where u differs by a fixed amount is increasing. Thus 
eradients of u are decreasing. A more concrete example 
of such an expansion wave is given in the next section. 
If the given values of wu are such that the straight 
characteristics converge, there is a compression wave 
in the region (see Fig. 2) and the gradients of w are 
increasing. 
Envelope of the Characteristics. Figure 2 shows that 
a family of converging straight lines defines a double 
