Wave shoaling and the angular frequency is proportional to its local rate of change, Simplifying to one dimension and cross-differentiating it is now easily seen that the above definitions indicate simply that the rate of change of wavenumber is balanced by the convergence of the frequency along a ray; Assuming stationary conditions (formula_27), this implies that wave crests are conserved and the frequency must remain constant along a wave ray as formula_28. As waves enter shallower waters, the decrease in group velocity caused by the reduction in water depth leads to a reduction in wave length formula_29 because the nondispersive shallow water
Distance oracle auxiliary data structure. A simple data structure that achieves this goal is a "matrix" which specifies, for each pair of nodes, the distance between them. This structure allows us to answer queries in constant time formula_2, but requires formula_3 extra space. It can be initialized in time formula_4 using an all-pairs shortest paths algorithm, such as the Floyd–Warshall algorithm. A DO lies between these two extremes. It uses less than formula_3 space in order to answer queries in less than formula_1 time. Most DOs have to compromise on accuracy, i.e. they don't return the accurate distance but rather a constant-factor