Consider 2-D, steady, inviscid, fully supersonic flow.
We wish to solve the equation
We could simply place points throughout the nozzle
area and solve them randomly. However, the beauty of the method of characteristics
is that it will tell us where to place our points.
y|
x------->
We now have 3 equations relating the disturbance potential j and its derivatives with the coordinate system (x,y). We need to solve for the 2nd derivative terms.
To accomplish this, we apply Cramer’s Rule. Let’s
first solve for 
Now, we can choose dx and dy to be any values, but they would not be most efficient. There is one solution where dx and dy will make D=0. So that we have a unique solution, we can choose the values u ,u , du, du such that u =0 as well.
Physically what does this mean?
So, in each direction, x & Y, there is a point
where 
cannot
be computed (not defined). However, the slopes 
can be defined. So the matrix system is indeterminant.
Along a stream line at pt A, there is a direction V where the velocities are indeterminant. Find, V and you’ve picked the best direction.
To do this, we solve D= 0.
divide through by 
use quadratic equation:
nu = Vcos qn= Vsin q
u
Close to A, the characteristic lines are the mach lines. However, the flow @A. inside the nozzle is interacting with other points, so the characteristic lines are curved.
(M is not constant)
Why are these points so important?
-Replace them into our numerator N=0
