Locally Integrable (f(x+ct) + f(x-ct))/2 is Weak Solution to Wave Equation

Theorem
Consider the equation of wave motion:


 * $\dfrac {\partial^2 u} {\partial t^2} - c^2 \dfrac {\partial^2 u} {\partial x^2} = 0$

with the initial conditions:


 * $\map u {x, 0} = \map f x$


 * $\map {\dfrac {\partial u} {\partial t}} {x, 0} = 0$

and $c \in \R$.

Then it has a weak solution of the form:


 * $\map u {x, t} := \dfrac {\map f {x + ct} + \map f {x - ct}} 2$

where $f \in \map {L^1_{loc} } \R$ is a locally integrable function.

Proof
Let $\map u {x, t} = \map f {x + ct}$ be a locally integrable function.

We have that a locally integrable function defines a distribution.

Let $T_u \in \map {\DD'} {\R^2}$ be a distribution associated with $u$.

Let $\phi \in \map \DD {\R^2}$ be a test function.

Then:

Therefore, $\ds \map u {x,t} := \frac {\map f {x + ct} + \map f {x - ct}} 2$ is a weak solution to the equation of wave motion.