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characteristic variables

ξ = x + ct
(6.3)
η = x − ct

this equation can be rewritten in the first canonical form, which is

u ξη = 0. (6.4)

To see that (6.3) is equivalent to (6.1), let us compute the partial deriva-
tives of u with respect to x and t in the new variables using the chain

rule.
u = cu − cu ,
t
ξ
η
u = u + u .
η
ξ
x
We can differentiate the above first order partial derivatives with re-
spect to t, respectively x using the chain rule again, to get
2
2
2
u = c u − 2c u + c u ,
ξξ
ηη
ξη
tt
u xx = u + 2u + u .
ηη
ξξ
ξη
Substituting this expressions into the left side of equation (6.1), we get
2
2
2
2
2
2
u −c u xx = c u −2c u +c u −c (u +2u +u ) = −4c u ξη = 0,
ξη
ξξ
ηη
tt
ξξ
ξη
ηη
which is equivalent to (6.4).
Equation (6.4) can be treated as a pair of two successive ODEs.
Integrating first with respect to the variable η, and then with respect
to ξ, we arrive at the solution
u(ξ, η) = f(ξ) + g(η) or u(x, y) = f(x + ct) + g(x − ct).
6.2 Initial Value Problem
Consider the following initial-value problem:


2
 u tt = c u xx , x ∈ R
u(x, 0) = φ(x) (6.5)

u t (x, 0) = ψ(x)
As should be familiar from ODE theory, we need to prescribe two pieces of initial data
to hope to get a unique solution. It is known that

u(x, t) = f(x + ct) + g(x − ct)

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