Forward dynamical problem for the Timoshenko beam

The initial boundary value problem

$$ \begin{array}{*{20}{c}} {\rho {u_{tt}} - {{\left( {\Gamma {u_x}} \right)}_x} + A{u_x} + Bu = 0,} \hfill & {x > 0,\quad 0 < t < T,} \hfill \\ {u\left| {_{t = 0}} \right. = {u_t}\left| {_{t = 0}} \right. = 0,} \hfill & {x \geq 0,} \hfill \\ {u\left| {_{x = 0}} \right. = f,} \hfill & {0 \leq t \leq T,} \hfill \\ \end{array} $$

is considered, where ρ = diag{ρ ₁, ρ ₂}, Γ = diag{γ ₁, γ ₂}, A, and B are smooth 2×2 matrix functions of x, whereas ρ _i and γ _i are smooth positive functions, provided that $0 < \frac{{{\rho_1}(x)}}{{{\gamma_1}(x)}} < \frac{{{\rho_2}(x)}}{{{\gamma_2}(x)}}$, x ≥ 0; f = col{f ₁(t), f ₂(t)} is a boundary control; u = u ^f (x, t) = col{u ₁ ^f (x, t), u ₂ ^f (x, t)} is a solution (wave). Such a problem describes wave processes in a system, where two different wave modes occur and propagate with different velocities. The modes interact, which implies interesting physical effects but, on the other hand, complicates the picture of waves.

For controls $f \in {L_2}\left( {\left( {0,T} \right);{\mathbb{R}^2}} \right)$, the problem is reduced to a relevant integral equation, generalized solutions u ^f are defined, and the well-possedness of the problem is established. Also, the fundamental matrix-valued solution is introduced and its leading singularities are studied.

The existence of “slow waves”, which are a certain mixture of modes that propagate with slow mode velocity, is established. Bibliography: 11 titles.