I am trying to numerically solve the equation for a inextensible, unshearable, Cosserat rod subject to planar bending moments. The ends of the rod are subject to force-free, moment-free BCs. The governing eqations are
$\rho I(s) \theta(s,t)_{tt} = [EI(s) \theta(s,t)_s]_s - [I(s)\sigma(s,t)]_s\,,\,\, t \geq 0\,,\,\, 0\leq s \leq L$
$\theta(s,0) = \theta(s,0)_t = 0$
$\theta(0,t)_s = \theta(1,t)_s = 0$
where $\sigma(0,t) = \sigma(L,t) = 0$. I can rewrite this system as
$\theta(s,t)_t = \omega_s$
$\rho I(s) \omega(s,t)_{st} = [EI(s) \theta(s,t)_s]_s - [I(s)\sigma(s,t)]_s\,\,\,\,\,$ (1)
where I can integrate (1) w.r.t. $s$:
$\rho \omega(s,t)_{t} = EI(s) \theta(s,t)_s - \sigma(s,t)$
I can further rewrite the system in terms of its characteristic variables
$c = \sqrt{E/\rho}$
$\eta = c \theta + \omega$
$\xi = c \theta - \omega$
$\eta_t = c \eta_s - \sigma$
$\xi_t = - c \xi_s + \sigma$
Now it is clear that the solution of $\eta$ is a leftward traveling wave and the solution of $\xi$ is a rightward traveling wave. Thus, $\eta$ needs a BC at $s=L$ and $\xi$ needs a BC at $s=0$. The general form of the boundary conditions is
$\eta(L,t) = \gamma_1 \xi(L,t)\,\,\,\,\,$ (2)
$\eta(0,t) = \gamma_2 \xi(0,t)\,\,\,\,\,$ (3)
I am very confused on how to implement the original BCs in terms of (2) and (3) (even after I take a derivative w.r.t. $s$). I'd appreciate it if someone can explain how to implement the right BCs for this problem.
Also, I am curious why I cannot reformulate the original equation as
$\theta(s,t)_t = \omega$
$\rho I(s) \omega(s,t)_{t} = [EI(s) \theta(s,t)_s]_s - [I(s)\sigma(s,t)]_s $
This form no longer looks hyperbolic, so I feel like I may be doing something wrong if I naively discritize this system and try to numerically implement it.