Periodicity

 

There are three common ways of quantifying periodic behavior in time. (When we begin to look at wave motion, we shall develop directly analogous quantities to describe periodicity in space.):

• Period T: The time for a full period or cycle. The period is typically measured in seconds, so that the basic unit is .
• Frequency f: The number of cycles which occur in a unit time. The unit of frequency is typically , which is defined as a special unit, ( ).
• Angular frequency : The number of radians of phase which pass in a unit time, where we associate radians of phase with each cycle. The unit of angular frequency is thus typically . Because radians carry no dimension ( is the ratio of the circumference of a circle to the radius, both measured in meters thus resulting in canceling units.), we frequently omit the radian and write the unit of angular frequency as simply .

All three of these measures describe basically the same thing and can be converted among each other. We shall use them interchangeable, and it thus is important that you be able to convert among them quickly using the following conversions.

If one cycle takes time T, then the number of cycles which occur per unit of time is

 

Because there are radians in one cycle, the number of radians which pass per unit time are therefore

 

where we remind ourselves that we may drop the optional ``unit'' . Finally, combining (1-2), we have