For the 1-D 2nd order wave eqn:
$$\frac{d^2u}{dt^2}=c\frac{d^2u}{dx^2}$$
subject to the boundary conditions: $$u(0,t)=0$$ $$\frac{du}{dx}(L,t)=C$$
where $C$ is some finite non-zero constant independent of time, is there an analytical solution to this problem?
Also subject to the initial conditions: $$u(x,0)=0$$ $$\frac{du}{dt}(x,0)=0$$
One can find the solution for this, using Separation of Variables, to be the following:
\begin{align} u(x,t) &= Cx - \sum_{m \in \mathbb{N}, \text{$m$ odd}} \frac{8 C L}{(m \pi)^2} \cos \left(\frac{m \pi}{2L}\left(L - x\right)\right) \cos \left( \frac{m\pi c^{1/2}}{2L}t\right) \end{align}
Note that the trick is what is mentioned in the comments, which is to do a change of variables using:
\begin{align} u(x,t) &= Cx+v(x,t) \end{align}
and then solving for $v(x,t)$, using the new initial conditions and simpler boundary conditions. With that, you can more easily construct the solution for $u(x,t)$.
To show correctness, a plot below is shown below comparing the above analytical solution with the numerical one. This was done using $L = 10$, $c^{1/2} = 5.8564 \cdot 10^{3}$, and $C = 3.7143 \cdot 10^{-6}$.
Additionally, below is a plot showing the $u(x,t)$ for a set of time values:
