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
Champernowne constant Champernowne constant In mathematics, the Champernowne constant is a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933. For base 10, the number is defined by concatenating representations of successive integers: Champernowne constants can also be constructed in other bases, similarly, for example: The Champernowne constants can be expressed exactly as infinite series: where formula_2 ceiling(formula_3), formula_4 in base 10, formula_5 and formula_6 is the base of the constant. A slightly different expression is given by Eric W. Weisstein (MathWorld): where formula_8