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15
1.4.3
The wave equation
u tt (x, t)= u xx (x, t)
arises in for example modeling the motion of a uniform string; see Wein-
berger [28]. Here, we want to solve the Cauchy problem for the wave
7
equation, i.e. (1.33) with initial data
u(x, 0) = φ(x)
(1.34)
and The Wave Equation 1.4 Cauchy Problems (1.33)
u t (x, 0) = ψ(x). (1.35)
But let us first concentrate on the equation (1.33) and derive possible
solutions of this equation. To this end, we introduce the new variables
ξ = x + t and η = x − t,
and define the function
v(ξ, η)= u(x, t). (1.36)
By the chain rule, we get
∂ð ∂η
u x = v ξ + v η = v ξ + v η
∂x ∂x
and

u xx = v ξξ +2v ξη + v ηη .
Similarly, we have

u tt = v ξξ − 2v ξη + v ηη ,
and thus (1.33) implies that

0= u tt − u xx = −4v ξη .

Since
v ξη = 0 (1.37)

we easily see that
v(ξ, η)= f(ξ)+ g(η). (1.38)

7 Initial-boundary value problems for the wave equation are studied in Chapter 5.