Variations of amplitude, wavelength and phase velocity of plane progressing waves over uniformly sloping beaches

The mysterious variation of the amplitude of waves progressing towards the shore over sloping beaches has been clarified explicitly by linear theory. The phase velocity of the waves increases rapidly as the waves travel across the surf zone towards the shore, as does the wavelength. The phase of the waves near shore is nearly stationary if the beach slope is small. Hence the waves are not sinusoidal near shore but surge towards the shore if the slope is small.