Characteristic coordinates are coordinates so that a' and c' vanish:
Finding characteristic coordinates:
Vanishing of a' requires that satisfies
To solve the equation for (
goes the same way), divide by
:
By taking the other sign for the square root, you can get a second
independent coordinate .
Bottom line, to get characteristic coordinates, solve the plus and minus
sign ODE above, and equate the integration constants to and
.
Notes:
Application to the wave equation:
utt - a2 uxx = 0
Since d' remains zero:
u = f1(x-at)+f2(x+at),
which is a right travelling 'wave' plus a left travelling one. Example 4.10 figures out what f1 and f2 are in terms of given initial displacement u and velocity ut at the initial time.