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WAVES

In a purely mathematical approach to the phenomenology of waves, we might choose to start with the WAVE EQUATION, a differential equation describing the qualitative features of wave propagation in the same way that SHM is characterized by   $\ddot{x} = - \omega^2 x$. The advantage of such an approach is that one gains confidence that any phenomenon that can be shown to obey the WAVE EQUATION will automatically exhibit all the characteristic properties of wave motion. This is a very economical way of looking at things.

Unfortunately, the phenomenology of wave motion is not very familiar to most beginners - at least not in the mathematical form we will need here; so in this instance I will adopt the approach used in most first year Physics textbooks for almost everything: I will start with the answer (the simplest solution to the WAVE EQUATION) and explore its properties before proceeding to show that it is indeed a solution of the WAVE EQUATION - or, for that matter, before explaining what the WAVE EQUATION is.