Solve for $\phi:[0,L]\times [0,\infty) \to \mathbb R$:
\begin{align*} \frac{\partial^2}{\partial t^2} \phi(x,t) &= c^2\frac{\partial^2}{\partial x^2} \phi(x,t) \\ \frac{\partial^2}{\partial t^2} \phi(L,t) &= -r \frac{\partial}{\partial x} \phi(L,t) \\ \frac{\partial^2}{\partial t^2} \phi(0,t) &= \boxed{\kappa_h \frac{\partial^2}{\partial t\partial x} \phi(0,t)} + \kappa_p \frac{\partial}{\partial x} \phi(0,t) \\ \phi(x,0) &= f(x) \\ \frac{\partial}{\partial t} \phi(x,0) &= g(x) \end{align*} If the boxed term is missing, then $\partial\phi/\partial x$ satisfies Robin boundary conditions. With the boxed term, using d'Alembert's method, I can show that there are solutions of the form $$ \phi_n(x,t) = A_n \exp(w_n(t+x/c)) + B_n \exp(w_n(t-x/c)) ,$$ where $\text{Re}(w_n) < 0$ or $w_n = 0$. (This is assuming $r, \kappa_h, \kappa_p > 0$.) The solution is: \begin{equation} (\kappa_p + (\kappa_h - c) w_n)A_n = (\kappa_p + (\kappa_h + c) w_n) B_n \end{equation} \begin{equation} \tanh(L w_n/c) = -\frac{c w_n(r + \kappa_p + \kappa_h w_n)}{r \kappa_p + r \kappa_h w_n + c^2 w_n^2} \end{equation} For example, if $L = c = r = \kappa_d = \kappa_h = 1$, then using Mathematica the first few values of $w_n$ are: $0$, $-0.39382\pm1.41436i$, $-1.00138\pm4.09326 i$, $-1.30874\pm7.14689 i$, $-1.49908\pm10.2568 i$, $-1.63641\pm13.3829 i$, $-1.74384\pm16.5156 i$, $-1.83211\pm19.6515 i$.
Is it possible to show the set $\{\phi_n(x,0)\}$ are complete, that is, do they span $L_2[0,L]$? And can one decompose $f(x)$ and $g(x)$ into these $\phi_n(x,0)$ using something analogous to the Fourier coefficient formulas? I already checked numerically, and it looks like the $\phi_n(x,0)$ are not orthogonal to each other.
Is there a general theory for this kind of stuff? Even a reference would be very helpful.