{\cal T}.$$ The wave equation and the initial data imply on ${\cal S}$ the relations $$\label{A-consist-cond}
\partial_t^{2\,k}\,\phi= \Delta^k\,\phi_0 + 
\sum_{l = 0}^{k - 1} \Delta^l\,\partial_t^{2(k - 1 + l)} f|_{t = 0},$$ $$\partial_t^{2\,k + 1}\phi = \Delta^k\,\phi_1 + 
\sum_{l = 0}^{k - 1} \Delta^l\,\partial_t^{2(k - 1 + l) + 1} f|_{t = 0},
\quad \quad k = 0, 1, 2, \ldots,$$ where $\Delta$ denotes the flat Laplacian. The boundary condition implies on $\Sigma$ $$\label{B-consist-cond}
\sum_{j = 0}^k {k \choose j}\left\{
\partial_t^jB^t\,\partial^{k - j + 1}_t\,\phi + 
\sum_{a = 1}^3\partial_t^jB^a\,(\partial^{k - j}_t\,\phi)_{,\,a}\right\}
= \partial^{k}_t\,h,
\quad \quad k = 0, 1, 2, \ldots.$$ For our initial boundary value problem to admit smooth solutions, conditions (\[A-consist-cond\]) and (\[B-consist-cond\]), read for given function $f$ as conditions on the data $\phi_0$, $\phi_1$, and $h$, must be satisfied on $\Sigma$. We consider two special situations.

i\) The intended applications requires a fixed boundary datum $h$ and thus fixed values of $\partial^k_t \,h$ on $\Sigma$. Inserting the time derivatives of $\phi$ given by (\[A-consist-cond\]) in the relations (\[B-consist-cond\]) then results in a sequence of conditions on the space derivatives of the data $\phi_0$ and $\phi_1$ on $\Sigma$. Depending on the function $e$ the construction of such data 