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
Acoustic wave equation include acoustic wave equations that incorporate fractional derivative terms, see also the acoustic attenuation article or the survey paper. The wave equation describing sound in one dimension (position formula_3) is where formula_1 is the acoustic pressure (the local deviation from the ambient pressure), and where formula_6 is the speed of sound. Provided that the speed formula_6 is a constant, not dependent on frequency (the dispersionless case), then the most general solution is where formula_9 and formula_10 are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one (formula_9) travelling up the x-axis