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# [Phys-l] frequency: a modest proposal

Greetings all,

I've always been mildly disturbed by the definition of the unit hertz as equivalent to 1/second. (Such as can be found, for example, at <http://physics.nist.gov/cuu/Units/units.html> and <http://physics.nist.gov/cuu/Units/SIdiagram.html>.

What is missing from these web pages is any consideration of the concept of "phase." Here are some propositions with which I think everyone will agree...

(1) The unit 'radian' is just a special name given to the number 1, when used for measuring plane angles. (This is straight off the NIST sites.)
(2) Phase is properly measured in units such as degrees or radians. That is, phase is measured with the same units as plane angles.
(3) Phase therefore has the same physical dimension as plane angles.
(4) The rate of change of phase for a periodic signal can be properly expressed in units of radian/s. This is usually called "angular frequency".
(5) Combining (1) and (4), angular frequency can properly be expressed in units of 1/s.
(6) A periodic signal with and angular frequency of 1 s^{-1} does not have a frequency of 1 Hz. I'd like to have a more specific name for that second quantity; it seems like either "cyclic frequency" or "counting frequency" would do nicely. "Counting frequency" extends nicely to the concept of frequency of discrete objects.

Given these facts, it seems that one is faced with two choices:
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(A) Declare that angular frequency and counting frequency are two separate concepts, which just happen to be measured in the same units. In this view, it is fine to have a single signal that has an angular frequency of 2pi/s and a counting frequency of 1/s.

It seems that this is essentially the choice currently in vogue. And I guess it is not too outlandish; there is already the example of activity (SI unit: becquerel) as yet another separate concept which shares the same unit.
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(B) BUT, the two frequencies seem very closely linked. Option (A) is almost like declaring length and width to be separate concepts. (Of course, length and width aren't separate because of the rotation operation. So that analogy isn't perfect..)

So why not define the unit 'cycle' = 2pi radians, and then make
1 Hz = 1 cycle/s = 2pi radians/s = 2pi/s ?

Then, periodic signals would simply have one physical characteristic, 'frequency,' which could be expressed in either unit. The equation omega=2pi*f becomes a simple unit conversion, with units of radians/cycle on the 2pi. This very naturally extends the concept of phase to periodic non-sinusoidal signals, which I'm not sure is the case currently.

For the most part, I don't think that this would necessitate changes in how business is currently conducted. Counting frequency calculations would remain the same (using Hz), and angular frequency calculations would remain the same. It is just that a very natural link is added between them.

The one change I can think of is that the proper base unit for wavelength (as we now know it) would become meter/cycle. It would be strictly proper to write lambda = 10 m/cyc = 1.59 m, and also the wave number could be defined as k = 1/lambda. Those might look like big departures from current practice, but actually people would just never refer to wavelength in pure meters, and the wavenumber definition would be written k=(2pi rad/cycle)/lambda.

Can anyone see any problems with this second option, other than the near impossibility of changing tradition?

Cheers,
-- James
--
Dr. James McLean phone: (585) 245-5897
Dept. of Physics and Astronomy FAX: (585) 245-5116
SUNY Geneseo email: mclean@geneseo.edu
1 College Circle web: http://www.geneseo.edu/~mclean
Geneseo, NY 14454-1401