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, 
, that is represented by a sinusoidal function of both space and time,
![\[\eta(x,t)=Acos(kx-\omega t+\Phi), \]](https://uw.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-c2db9fd57bb83dbc35bd488de61289fa_l3.png "Rendered by QuickLaTeX.com")
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),
![\[ k=\frac{2 \pi}{\lambda} \quad \omega=\frac{2 \pi}{T}. \]](https://uw.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-c1822ba5bd1ead627dd29e8cd081c5c9_l3.png "Rendered by QuickLaTeX.com")
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,
![\[ C_{Phase} = \frac{\lambda}{T} = \frac{\omega}{k}. \]](https://uw.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-ae0a1d755408bf481d84f4bf2ac5bb94_l3.png "Rendered by QuickLaTeX.com")
Mini-lecture for class:
