This article aims to finalize the classification of weakly well-posed hyperbolic boundary value problems in the half-space. Such problems with loss of derivatives are rather classical in the literature and appear for example in [19] or [2]. It is known that depending on the kind of the area of the boundary of the frequency space on which the uniform Kreiss-Lopatinskii condition degenerates then the energy estimate can include different losses. The three first possible area of degeneracy have been studied in [9] and [3] by the use of geometric optics expansions. In this article we reiterate the same kind of tools in order to deal with the last remaining case, namely a degeneracy in the glancing area. In comparison to the first cases studied we will see that the equation giving the amplitude of the leading order in the expansion, and thus initializing the whole construction of the expansion, is not a transport equation any more but it is given by some Fourier multiplier. This multiplier need to be invert in order to recover the first amplitude. As an application we discuss the existing estimates of [11]-[10] for the wave equation with Neumann boundary condition.