A multiscale computational scheme is developed to use given small microscale simulations of complicated physical wave processes to empower macroscale system-level predictions. By coupling small patches of simulations over unsimulated space, large savings in computational time are realizable. Here, we generalize the patch scheme to the case of wave systems on staggered grids in two-dimensional (2D) space. Classic macroscale interpolation provides a generic coupling between patches that achieves consistency between the emergent macroscale simulation and the underlying microscale dynamics. Spectral analysis indicates that the resultant scheme empowers feasible computation of large macroscale simulations of wave systems even with complicated underlying physics. As an example of the scheme's application, we use it to simulate some simple scenarios of a given turbulent shallow water model.