Solution
= Solution
The material occupies $Y\geq0$, so its outward <unit normal> at the surface is $n=-e_Y$. A free surface has zero <traction>, $\tau n=0$. Therefore
$$
\boxed{\tau_{XY}(X,0,t)=0,\qquad\tau_{YY}(X,0,t)=0,\qquad\tau_{ZY}(X,0,t)=0.}
$$
The third condition is automatic for the two-dimensional <displacement field> with zero third component and no dependence on the third coordinate. The first two impose zero shear and normal <stress> at the free surface.