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Re: [Phys-l] frequency: a modest proposal

I must admit that I don't really see the problem with the current use of "frequency", "angular frequency", and "phase". This might be because I work in acoustics, and everyone uses these terms (and "phase angle") without much trouble, and understands that there is just a difference of multiplier between them. Which term one uses is based on whom one is talking to (non-technical visitors are much more comfortable with "f"). I will say that students first encountering the terms must be made to be careful, especially when ambiguity between "counting frequency" and "angular frequency" can arise.

We *always* use Hz to be equivalent to s^-1, to the extent that things like sound speeds are often written as "the speed of sound in water is approximately 1.485 mm MHz"; this makes the concepts of wavenumbers and wavelengths immediately apparent.

It seems to me that the analogy might be a little more apt if one uses "Avogadro's number", which is, after all, just a prefix for "lots and lots of things".

Down with categorical imperative!

From: James McLean <>
To: Physics List <>
Sent: Wed, January 27, 2010 1:33:53 AM
Subject: [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
<> and

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

(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:
(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.
(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

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?

-- James
Dr. James McLean phone: (585) 245-5897
Dept. of Physics and Astronomy FAX: (585) 245-5116
SUNY Geneseo email:
1 College Circle web:
Geneseo, NY 14454-1401
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