The effects connected with coincidence of velocities in the two-velocities dynamical system

The paper deals with the system
\begin{align*} &\rho u_{tt}-u_{xx}+Vu=0,\quad x>0,\quad t>0;\\ &u|_{t=0}=u_t|_{t=0}=0;\\ &u|_{x=0} = f, \end{align*}
where $\rho=\rho(x)$ and $V=V(x)$ are $2\times2$-matrix functions; $\rho=\operatorname{diag}\{\rho_1,\rho_2\},\rho_{\alpha}>0$; $f$ is a boundary control; $u=u(x,t)$ is the solution. The singularities of the fundamental solution corresponding to the controls $\binom{\delta}0$ and $\binom0{\delta}$ ($\delta=\delta(t)$ is the Dirac $\delta$-function) are under investigation. In the case of $\rho_1(x)\ne\rho_2(x)$ the singularities of the fundamental solution are described in terms of the standard scale $\delta,\int\delta, \iint\delta,\ldots$. In the presence of points $x=x_*:\rho_1(x_*)=\rho_2(x_*)$ an interesting effect occurs: the singularities of intermediate (fractional) orders appear.