For a great summary of the mathematical representation of wave propagation in one spatial dimension, see Dan Russell’s webpage Wave Motion in Time and Space. A wave propagating in one spatial dimension has a displacement from equilibrium, , that is represented by a sinusoidal function of both space and time,
where k is the wavenumber (inversely related to the wavelength, , the distance between wave crests) and is the frequency and (inversely related to the period of the oscillation, T),
The phase constant or phase shift, , determines the value of the displacement at t=0 and x=0. Whether we use a cosine or a sine function is arbitrary because one can be changed into the other by using a phase shift of .
Wave crests travel through space at a rate given by the phase speed,