Given the conversions in Section 3.2 and the conversions
among temporal quantities from the previous set of notes, we can
convert any temporal quantity into any other and any spatial quantity
into any other. What we lack is an ability to link one of the spatial
quantities into one of the temporal quantities. Then, we can convert
any of the six quantities T, f, 
, 
, 
, k
to any other. A relation linking a spatial and temporal quantity is
known as a dispersion relation.
To generate one such relation, consider the wave illustrated in Figure 1. Suppose that the wave travels to the right with wave speed v. The wave speed does not necessarily refer to the speed of individual parts of the string, but rather to the speed at which the pattern of the wave moves.
As the wave passes, the point at location 
first moves down into
the trough of the oncoming wave and then comes back up as the crest of
the next wave comes directly under the point. This up-down-up process
represents one full period and takes time T. During this time, the
crest of the oncoming wave has moved precisely one repeat distance, or
wavelength . The speed of the wave, wave speed v, is
therefore
As this relates one of the spatial and one of the temporal quantities together, Eq. 4 qualifies as a dispersion relation and allows us to convert among any of the six basic wave quantities provided that we know the value of the wave speed v. To determine this speed for the string, we now proceed to the detailed analysis of the motion of the string.
