Forward dynamical problem for Timoshenko beam

We deal with an initial boundary value problem of the form
\begin{align*} &\rho u_{tt}-(\Gamma u_x) _x+Au_x+Bu=0,\qquad x>0,\quad 0<t<T,\\ &u|_{t=0}=u_t|_{t=0}=0,\qquad x\geq0,\\ &u|_{x=0}=f,\qquad0\leq t\leq T, \end{align*}
where $\rho=\mathrm{diag}\{\rho_1,\rho_2\}$, $\Gamma=\mathrm{diag}\{\gamma_1,\gamma _2\}$, $A$, and $B$ are smooth $2\times2$-matrix functions of $x$, whereas $\rho_i,\gamma_i$ are smooth positive functions provided $0<\frac{\rho_1(x)}{\gamma_1(x)}<\frac{\rho_2(x)}{\gamma_2(x)}$, $x\geq0$; $f=\mathrm{col}\{f_1(t),f_2(t)\}$ is a boundary control; $u=u^f(x,t)=\mathrm{col}\{u_1^f(x,t),u_2^f(x,t)\}$ is a solution (wave). Such a problem describes the wave processes in a system, where two different wave modes occur and propagate with different velocities. The modes interact that implies interesting physical effects but, on the other hand, complicates the picture of waves. For controls $f\in L_2((0,T);\mathbb R^2)$, we reduce the problem to the relevant integral equation, define the the generalized solutions $u^f$, and establish the well-possedness of the problem. Also, the fundamental matrix-valued solution is introduced and its leading singularities are studied. The existence of the “slow waves” that are the certain mixture of modes, which propagate with the slow mode velocity, is established. Bibl. – 11 titles.