In this paper we make a comprehensive investigation of well-posed initial-boundary value problems for 2$\times$2 hyperbolic systems in a 2-dimensional half space domain ${\mathbb R}^2_+\equiv \{(x,y)|x>0, y \in {\mathbb R}\}$ and obtain their solutions in explicit form. We convert the boundary and the initial data into information on the boundary only so that the Master Relationship program introduced in [8, 9, 10] can be applied here and gives an intrinsic algebraic relationship in the boundary data in terms of their Fourier-Laplace transform variables. With the intrinsic algebraic relationship and the well-posed boundary conditions we can solve the full boundary data algebraically and explicitly in the Laplace-Fourier transformed variables. Those algebraic solutions give us the surface wave propagators. We then factorize the surface wave propagators into simple operators so that they can be inverted explicitly into the physical domain. This way we obtain an explicit formula for the solution of the initial boundary problem for the 2$\times$2 hyperbolic systems in the space-time domain; and, in particular, the Green's functions are constructed.

We also apply the same framework to the model wave equation $(\partial_t + \Lambda \partial_x)^2 u -\Delta u=0$ for the Dirichlet, Neumann, Robin's, Radiative, Absorbing boundary conditions in the half 2-dimensional space domain. Various interesting wave propagation structures corresponding to different boundary conditions are identified in terms of the determinants of the surface wave propagators.