41 Propagating waves (1D)

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, \eta, that is represented by a sinusoidal function of both space and time,

    \[\eta(x,t)=Acos(kx-\omega t+\Phi), \]

where k is the wavenumber (inversely related to the wavelength, \lambda, the distance between wave crests) and \omega is the frequency and (inversely related to the period of the oscillation, T),

    \[ k=\frac{2 \pi}{\lambda} \quad \omega=\frac{2 \pi}{T}. \]

The phase constant or phase shift, \Phi, 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 \pi/2.

Wave crests travel through space at a rate given by the phase speed,

    \[ C_{Phase} = \frac{\lambda}{T} = \frac{\omega}{k}. \]