Suppose $h(x,t)=\phi(x)+\psi(x,t)$ where $h(x,t)=h(x+L,t)$. What are the requirements on $\phi(x)$ and $\psi(x,t)$ in terms of their spatial periodicity? No special requirements on the temporal part of $h(x,t)$.
If either $\phi(x)$ or $\psi(x,t)$ follows the same spatial periodicity as $h(x,t)$, then it is easy to see that the other part must also be periodic. But what about the case when neither is periodic? Is it possible that both $\phi(x)$ and $\psi(x,t)$ are not spatially periodic but their sum $h(x,t)$ is spatially periodic? How to prove?
Let $\phi(x) = e^x + \sin(2 \pi x/L)$ and $\psi(x,t) = -e^x$. Then neither $\phi$ or $\psi$ is periodic, but $h(x,t) = \sin(2 \pi x/L)$ and $h(x+L,t) = h(x,t)$.