Harmonic Waves

As I hope we emphasized above, linear media can propagate any wave form we desire at the specific velocity which characterizes the medium. We shall now specialize in the propagation of a particularly convenient and periodic wave form namely a harmonic wave.

k is a constant of dimension inverse length which we shall return to below. There are two reasons for paying special attention to harmonic waves :

• Any function can be built up by a unique superposition of harmonic functions. This is the content of Fourier's theorem.
• harmonic functions propagate unaltered through dispersive as well as (of course) non-dispersive linear media.

We just finished talking about periodic phenomena as oscillations and they were characterized by a period, a frequency or a angular frequency. Multiplying out the argument to the sinus function we have :

We identify

as the angular frequency. Of course we still have the relations

where f now symbolizes the frequency not to be mistaken for the time and space dependent disturbance of the medium, f(x-vt) and T is the period not to be mistaken for the tension in a string! Very often we write

where it is understood that this wave can actually only propagate through a given medium if omega and k satisfy a ``dispersion relation'' as it is called :

k is the so-called wave number, the analogue of the angular frequency for the spatially periodic behavior. The repeat distance in space is

We have to be able to convert between all the different variables characterizing a traveling periodic wave remembering that in a non-dispersive medium there is a single possible velocity and hence the spatial periodicity fixes the temporal periodicity and vise versa :

 
• Interrelationship of spatial and temporal periods and frequencies
• Energy transport in Harmonic Waves