Consider an semi-infinite string stretched between $2$ fixed points. Let $u(x, t)$ be the displacement of a string, at position $x$ and time $t.$
We describe the wave equation by: $$u(x, t) = k_1(x − ct) + k_2(x + ct)$$ for arbitrary functions $k_1(z)$ and $k_2(z).$
The string is subject to boundary conditions: $$u(0, t) = u(1, t) = 0 ,\: t > 0.$$
The string has an initial displacement $u(x, 0) = f(x), x ∈ (0, 1)$ and is initially at rest.
Combine the derived expressions of $k_1(z)$ and $k_2(z)$, to deduce that $k_1(z)$ and $k_2(z)$ are periodic with period $2$.
For the derivation of the expressions, Ive already deduced that $k_1(z) = −k_2(−z)$ and $k_2(1+z)=-k_1(1-z)$. My problem is that I cant prove that this is periodic or that it has a period of $2$ so any help will be appreciated.
Since $k_1(z)=-k_2(-z)$, it suffices to show that $k_1$ is periodic with period $2$. With this in mind, let's go at it directly: $$ k_1(2+z) = k_1(1+(1+z)) = -k_2(1-(1+z)) = -k_2(-z) = k_1(z), $$ where we have used $k_1(1+w)=-k_2(1-w)$ and then $k_1(z)=-k_2(-z)$. So the period is at most $2$, but may be $2/n$ for some integer $n$.
In fact the period can't be less than $2$, but we need to lean on the initial conditions to do this: together they imply that $k_1(z)=k_2(z)=f(z)/2$ (see here), and so $$ k_1(1+z)=-k_2(1-z) = -f(1-z)/2 \neq f(z)/2 = k_1(z) $$ in general; moreover, the period can't be less than one unless $f$ is periodic.