The extension of vortex-wave interaction theory to generate a new kind of exact coherent structure is discussed. In an infinite uniform shear flow the new structure investigated is made up of an infinite array of waves propagating on a predominately unidirectional flow. The waves are inviscid except for critical layers where their singular nature is regularized by viscosity. Within each of the periodically spaced critical layers associated with the waves nonlinear effects drive a stress jump in a roll flow in a plane perpendicular to the predominant flow direction. The roll structure driven by those interactions is periodic in both directions perpendicular to the predominant flow direction and is also periodic in time. The wave system sustaining the interaction is periodic in all three spatial directions and time. It is shown that in a more general shear flow the structure can be used locally to describe similar states. If the critical layers are sufficiently distant then the periodic structure leads to the law of the wall which occurs in turbulent boundary layers. An asymptotic theory describing how the structure can be formally embedded into an arbitrary shear flow is given. Now the mean flow is sustained by the locally quadruply-periodic roll-streak-wave system but the condition that the local solutions can be embedded in a nonlocal slowly varying asymptotic theory in the spirit of ray theory shows that the mean state must satisfy a nonlinear differential equation. The possible relevance of the structures found from the equation to the logarithmically varying region of a turbulent shear flow is discussed. The results suggest that a weakly coupled array of exact coherent structures is consistent with the law of the wall but that as the coupling strengthens the mean flow can have an algebraic dependence on distance from the wall.