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

[John Denker <jsd@av8n.com>]

On 02/02/2010 01:09 AM, James McLean wrote:

> * If regularly spaced widgets are rolling off a conveyor belt, and you 
> /count/ how many come out in each second, then an appropriate unit is 
> "widgets/s" = Hz = 1/s.
>
> * If instead a square wave is rolling off the conveyor belt, then the 
> situation is directly analogous, where the "objects" being counted are 
> now repetitions of the waveform shape, called cycles.
> "cycles/s" = Hz = 1/s = widgets/s
>
> * Now it is a very small jump to sinusoidal waves.  

Direct hit!  That is a masterful pedagogical example.  I now
see that we are discussing a profoundly nontrivial dilemma:
 1) For sine waves, radian measure is conventional and
  convenient.
 2) For square waves, counting cycles is convenetional
  and convenient.

Continuing down that road, we find there is a third horn to 
this dilemma:
 3) For triangle waves, radians are sometimes more convenient
  than cycles ... and sometimes vice versa.

I should have seen this earlier ... but sometimes I need simple 
things to be explained to me.  

I have more than once been impaled on the triangle-wave horn
of this dilemma, e.g. when writing a subroutine that computes
a triangle waveform.  Things get even more miserable when we
we consider multipole distributions.  In two dimensions:
  3b) For a p-wave, the real part of the wavefunction goes
   like sin(θ) while the probability density goes like sin(2θ).
  3c) For a d-wave, the real part of the wavefunction goes
   like sin(2θ) while the probability density goes like sin(4θ).

... so what could we possibly mean by "the" period of such a
thing?  It's a dilemma, or worse.

> Thus we have two 
> different concepts (counting frequency and angular frequency) which can 
> be measured for the sinusoidal wave. 

Quite so.

> They share the same dimensions and 
> units, but they are distinct. Just as torque and energy share the same 
> dimensions and units, but are distinct.

Ooops, lost me there.  I would say that cycles and radians
have the same dimensions but *different* units.

If the units are the same it implies the dimensions are
the same, but not vice versa.
  http://www.av8n.com/physics/dimensionless-units.htm

> If we push my "modest proposal" to an extreme, to entirely get rid of 
> Hz=1/s, then I guess the base unit for counting objects would become the 
> radian.  5 widgets = 31.415 radians.  I have to admit, that's absurd. 

At this point I like to trot out the international "no extremism"
symbol:
  http://www.av8n.com/physics/img48/no-extremism.png

It seems obvious that worrying about "the" units or the "base"
units is a waste of time.  Holy war between the big-endians and
the little-endians.

It is reasonable to ask what units are convenient in this-or-that
situation.

> But I'm not wholly satisfied the the "two distinct units" solution either.

I reckon that's the only solution.  I can live with it.

Returning to the conveyor analogy:  Suppose the conveyor is
carrying cartons each containing a dozen eggs.  Do you count
cartons, or count eggs?

And if dozens aren't bad enough, what about moles?  In atomic
physics, "the" energy scale is kT ... while for macroscopic
systems RT is often more convenient.  For a spin-3/2 system,
should we say s = 4 bits per particle (entropy per particle)
or should we say s = R ln(4) (entropy per mole)?

And speaking of logarithms, I reckon there are a lot more
than two distinct units in common use:
 -- log base 10
 -- log base e
 -- log base 2
 -- log base 1.25893 i.e. dB
 -- et cetera.

======================

Bottom line constructive suggestion:  It pays to keep track
of the units ... even for dimensionless units.  Degrees are
different from radians are different from cycles.

A hertz should be defined as a cycle per second, not as an
inverse second.

I recommend that "inverse second" should not be used as a
unit of frequency, because it is ambiguous at best.  If you
mean cycles per second, say "cycles per second".