Periodic functions

We encounter periodic: phenomena every day of our lives. Those of us who still use analogue clocks are acutely aware of the 60 second, 60 minute and 12 hour periods associated with the sweeps of the second, minute and hour hands. We are conscious of the fact that the Earth rotates on its axis roughly every 24 hours and that it completes a revolution of the Sun roughly every 365 days. These periodicities are reasonably accurate. The quantities we are interested in measuring are not precisely periodic and there will also be error associated with their measurement. Indeed, some phenomena only seem periodic. For example, some biological population sizes appear to fluctuate regularly over a long period of time, but it is hard to justify using common sense any periodicity other than that associated with the annual cycle. It has been argued in the past that some cycles occur because of predator-prey interaction, while in other cases there is no obvious reason. On the other hand, the sound associated with musical instruments can reasonably be thought of as periodic, locally in time, since musical notes are produced by regular vibration and propagated through the air via the regular compression and expansion of the air. The ‘signal’ will not be exactly periodic, since there are errors associated with the production of the sound, with its transmission through the air (since the air is not a uniform medium) and because the ear is not a perfect receiver.