An interesting special case of this solution is the case of *harmonic* waves propagating to the left or right. Harmonic waves are
simply waves that oscillate with a given harmonic frequency. For
example:

(113) |

This particular wave looks like a sinusoidal wave propagating to the right (positive direction). But this is not a very convenient parameterization. To better describe a general harmonic wave, we need to introduce the following quantities:

- The
**frequency**. This is the number of cycles per second that pass a point or that a point on the string moves up and down. - The
**wavelength**. This the distance one has to travel down the string to return to the same point in the wave cycle at any given instant in time.

To convert (a distance) into an angle in radians, we need to
multiply it by radians per wavelength. We therefore define the
*wave number*:

(114) |

(115) | |||

(116) | |||

(117) |

where we have used the following train of algebra in the last step:

(118) |

(119) |

As before, you should simply *know* every relation in this set of
algebraic relations between
to save time on
tests and quizzes. Of course there is also the harmonic wave travelling
to the left as well:

(120) |

A final observation about these harmonic waves is that because arbitrary
functions can be *expanded* in terms of harmonic functions (e.g.
Fourier Series, Fourier Transforms) and because the wave equation is
linear and its solutions are superposable, the two solution forms above
are not really distinct. One can expand the ``arbitrary''
in a sum of
's for special frequencies and
wavelengths. In one dimension this doesn't give you much, but in two or
more dimensions this process helps one compute the *dispersion* of
the wave caused by the wave ``spreading out'' in multiple dimensions and
reducing its amplitude.