Characteristic Coordinates

Characteristic coordinates are coordinates so that a' and c' vanish:

Finding characteristic coordinates:

Vanishing of a' requires that satisfies

while for c' to vanish, must also satisfy this equation.

Bottom line:

Solve the differential equation

and take the integration constant to be . By taking the other sign for the root, you can get a second independent coordinate .

Notes:

1.

Since integration constants are not unique, the characteristic coordinates are not. But the lines of constant and are unique, and are called characteristic lines or characteristics.

2.

Elliptic equations do not have characteristics, and parabolic ones only a single family.

Application to the wave equation:

u_tt - a² u_xx = 0

Since d' remains zero:

D'Alembert solution:

u = f₁(x-at)+f₂(x+at),

which is a right travelling 'wave' plus a left travelling one.

11/03/00 0:09:07