[Geezer]

That's why we need to analyse the case that's pertinent to the original question of an object with mass being accelerated by a force. In that situation, the object has a cycle defined by 360 degrees of rotation. There can be no discontinuity in its rotation.

One or more cycles will differ from 24 hours. Therefore, there was a change in the period during some number of cycles. And you can't change the period without changing the frequency.

[Bored chemist]

"There can be no discontinuity in its rotation."
The rotation sped up and slowed down, it may have been continuous, but it wasn't steady either.
Anyway, as I have said before "international cow fart day" happened once. It doesn't have a frequency.
The frequency of rotation was always 1 in 24 hrs. What the cows did was slightly affect the phase.

[Geezer]

The frequency of rotation was always 1 in 24 hrs. What the cows did was slightly affect the phase.

Assuming we are talking about the Earth, the only way the phase could change without affecting the 24 hour cycle would be if the phase shifted in one direction, then shifted back an equal amount in the other direction, and both shifts happened within a 24 hour cycle. Otherwise, the rotation of at least one cycle would have to deviate from 24 hours. Are you saying that did not happen? I think you will have to accept that it actually did.

Knowing that, within some number of cycles that includes the ones that deviated from 24 hours, if we sum all their cycle times and divide the result by the number of cycles to get the period, the result is not going to be 24 hours. We might pick a very large number of cycles to minimize the apparent deviation from 24 hours, but unless we pick an infinite number of cycles, there will always be a deviation.

The only other way to avoid the change in period is to ignore any cycles that deviate from 24 hours.

Your "working day" model is interesting, but it has a teensy flaw. Your "clock" frequency is the same before and after the phase shift. But you make no mention of what happened to your clock during the phase shift.

Let's assume your clock is the alarm clock that wakes you up to go to work. In that case, the phase shift occurred while you were adjusting the time that the alarm goes off. The rate at which you adjusted the wake-up time determines the spectrum of frequencies produced during the phase shift.

For example, if you adjusted the wake-up time in zero time, you would generate an infinite series of frequencies. That might make it make it a bit difficult to claim there was no change in frequency.   

[damocles]

Something that happens once, and only once doesn't have a frequency. It doesn't matter how long it takes, Only things that repeat have a meaningful frequency.
My granny may have lived for a hundred years, but she didn't live ten times per millennium.

And I'm sorry you don't understand that my working day and the earth's rotation have a simple fixed phase relation, except once when I changed it.
I used to get up at 08:30, that's about 60 degrees of the earth's rotation before the Sun is overhead. Now I get up at 06:30; that's about 90 degrees before noon. (I'm ignoring the half hours to keep the arithmetic easy.)
The change of 30 degrees is a real phase shift.
(obviously I'm simplifying it by also ignoring weekends, BST, and such)

One of the important things about my redesign of the traffic scheme is that it is an ongoing effect -- it will last as long as our civilization or fossil fuel reserves do. It is not a constant effect, so we would have to talk about an increase in average frequency, but it would constitute a real frequency shift rather than just a phase shift.

Similarly with the cows -- if they were lined up for an event, then sure, the net effect is only a phase shift. But if they were to stay lined up, and someone were to conduct an ongoing symphony, then there would be a frequency shift.

[Geezer]

but it would constitute a real frequency shift rather than just a phase shift.

Not to be picky, but any phase shift produces a real change in frequency (it might be small, but it's still real.)

Please see my previous post - I was adding some stuff to it while you were posting - possibly a "no-no", but I didn't think anyone else was actually awake []

[Bored chemist]

" but unless we pick an infinite number of cycles"
At last! You have got it! A single one-off event in the whole of time doesn't produce a frequency shift.
Incidentally, any other number of cycles would be totally arbitrary and hard to justify.

We can stop now.

[Geezer]

A single one-off event in the whole of time doesn't produce a frequency shift.

It's not a one-off event. As you pointed out yourself, it's a phase shift, and we can't change phase without changing frequency.

It might be a one-off event if the Earth instantaneously rotated through some number of degrees, but as that is clearly impossible, we don't need to worry about it.

We can stop now.

[JP]

Just to throw a monkey into the butter:

Since the frequency spectrum is the Fourier transform of the signal, terms like "frequency change" are somewhat problematic.  If I do a Fourier transform of your signal, I get a frequency spectrum for the entire signal, including the phase change.  This frequency doesn't change over time.

By the way, I worked out the Fourier transform for a sinusoidal signal with an instantaneous phase change and it gives you the same frequency components associated with a single sinusoid. 

If the phase change is gradual, it will generate more frequency components.

---------------------

If you do want to talk about frequency changing over time, I suggest using a spectrogram, in which case even an instantaneous phase change causes a frequency shift in the spectrogram.

Can we stop now?

[Geezer]

Since the frequency spectrum is the Fourier transform of the signal, terms like "frequency change" are somewhat problematic. 

It's only problematic if you assume "frequency" implies sinusoidal.

I think we can stop now.

[JP]

True.  It's probably best to agree on a definition of frequency before arguing over specific examples.

You have shamed me into stopping.

For now.

[Bored chemist]

It's clear that Geezer and I have differing opinions of the meaning of frequency.
I think that something needs to repeat before it has a meaningful frequency.
Sure, you can FT a single spike or top hat, but what you get depends on the apodisation. If you don't arbitrarily crop the time domain then you get a zero frequency. If you accept that the input function is infinitely wide you don't get a spectrum.

Of course, if you could get the cows to line up and fart regularly, say every Tuesday, that would be different.

[Geezer]

It's clear that Geezer and I have differing opinions of the meaning of frequency.
I think that something needs to repeat before it has a meaningful frequency.
Sure, you can FT a single spike or top hat, but what you get depends on the apodisation. If you don't arbitrarily crop the time domain then you get a zero frequency. If you accept that the input function is infinitely wide you don't get a spectrum.

Of course, if you could get the cows to line up and fart regularly, say every Tuesday, that would be different.

Aha! Now we are getting somewhere.

I can think of three cases. There may be more.

a) A purely sinusoidal "frequency". You can't change its phase at all.

b) A non-sinusoidal "frequency". You can change its phase, but not without changing its "frequency".

c) A non-sinusoidal function composed of more than one sinusoidal "frequency". You can change the phase of the function by altering the amplitudes of the sinusoidal frequencies.

[JP]

Geezer, what's the definition of frequency that gives you the above three cases?  I'm betting that if you give a definition, we can work out pretty quickly why BC's definition doesn't agree with it (and for that matter, why the Fourier transform definition doesn't agree with either).

[Geezer]

The one in Wikipedia seems reasonable enough.

"Frequency is the number of occurrences of a repeating event per unit time."

I think a repeating event could be lots of different things, but two ought to be sufficient. The unit of time could be anything you like.

[JP]

Ok, what about the cow example?  I'll make up some numbers, so don't complain if they're quite a bit off. 

Let's say the earth rotates 1 time/24 hours.  One afternoon, and only once, the cows all fart and that day is shortened by 6 hours.  When the sun rises again and all following day, the day is once again 24 hours long.  So you have days of 24 hours, one day of 18 hours, followed by days of 24 hours.

What would your definition tell us about the frequency of the earth's rotation in this case?  If you need more information, feel free to specify how the earth speeds up for that one day, for example.

[Geezer]

Ok, what about the cow example?  I'll make up some numbers, so don't complain if they're quite a bit off. 

Let's say the earth rotates 1 time/24 hours.  One afternoon, and only once, the cows all fart and that day is shortened by 6 hours.  When the sun rises again and all following day, the day is once again 24 hours long.  So you have days of 24 hours, one day of 18 hours, followed by days of 24 hours.

What would your definition tell us about the frequency of the earth's rotation in this case?  If you need more information, feel free to specify how the earth speeds up for that one day, for example.

That would not work. We'd end up shortening a lot of days if the cows knocked six hours off one of them. I would think the retarding torque produced by atmospheric friction as a result of the cow exhaust would be rather small, and it would probably decay exponentially. In the interest of keeping it as simple as possible, let's make that assumption. I'm also assuming the moment of inertia of the cow/earth system is constant, even although it would not be.

I'd suggest we assume that, after a (an?) heroic effort, the cows manage to whack a whole second off one day. Although I think it's pretty unlikely, that way we might assume the Earth's angular velocity returned to its previous value in less than 360 degrees.

Again, in the interests of not making it too complicated, let's assume there was constant angular acceleration for one hour sufficient to knock one second off a day, followed by constant angular deceleration for 11 hours, after which the angular velocity returned to 360 degrees/day. (Assuming a "day" is determined by an atomic clock.)

Applying the Wiki definition, and assuming the unit of time is one day, we have a frequency of about 1.0000115 cycles/day.

Personally, I'd prefer to define the Earth's frequency using something with a lot more resolution like, for instance, arcseconds per second (should be around 15 Hz), as that would make the frequency change much more apparent, but it really won't make any difference as long as we understand the angular acceleration.

(BTW, if you do use arcseconds per second, the average frequency during our "short day" is 15.000173 Hz)   

[JP]

"Frequency is the number of occurrences of a repeating event per unit time."

So we need an event to repeat.  I'm guessing here, but is your repeating event one sunrise or sunset (or a 2*pi radian rotation of the earth)?  If so, you can get frequency by calculating (# of radians rotated through)/(# of seconds). 

If you're doing it this way, you have to specify two things:
1) How many radians or seconds are you averaging over?
2) What is the starting time of your averaging?

For your case, I believe it was 2*pi radians and you started right when the cows farted?

[Geezer]

For your case, I believe it was 2*pi radians and you started right when the cows farted?

That would work.

[JP]

For your case, I believe it was 2*pi radians and you started right when the cows farted?

That would work.

Ok, I understand your definition and agree that with that definition, you get a changing frequency. 

I think that BC's definition of a repeating event was that the event (one rotation of the earth through 360 degrees) had to be identical in duration to the other events to be considered repeating, though he can correct me if I'm wrong.  So in that case, you can't even think of frequency defined over any interval that includes the short day.

Your definition is that it the event is rotation through 360 degrees, but in terms of repetition, you count it as repeating so long as the earth keeps rotating, even if the time it takes for each rotation varies. 

The Wiki definition of frequency is pretty vague, so I think both of those interpretations make sense within its bounds. 

For whatever it's worth, my own preferred definition of frequency would be in terms of the Fourier transform, which has aspects of both of these definitions. 

[Bored chemist]

FFS!
International cow fart day happens just once.

Not once a day.
Not twice, separated by  some time period (of any duration you might want to choose).
Not alternate Tuesdays.
Not every full moon.
Once. Only once just on one occasion.
Once.
It doesn't repeat.
It has no repetition.
Since it happens once, and never again, it does not happen more than once.

Am I beginning to get the message through here?
Since, as wiki says, ""Frequency is the number of occurrences of a repeating event per unit time."" and we are not talking about a repeating event, it doesn't have a frequency.
Ok, now to look at a nice easy case of the maths.
Imagine it's not cows and  farts but some huge gun and it launches a big cannon ball but the ball is hooked to a long chain which is nailed to the earth.
The gun fires and pushes the earth.
After a while the chain pulls tight and it then tugs  on the earth. The earth is initially slowed down, but the tug on the chain pulls it back to it's original speed.

It's a really big fast ball, and a really strong long chain.
So big that when the gun fires, it stops the earth dead. (OK you would need two on opposite sides of the earth to get a torque). 6 hrs later the chain goes tight an pulls the world back to spinning again- once every 24 hrs, but 6 hours late- a rather big phase shift of 6/24 *360 ie 90 degrees.

Not only does this cause absolute mayhem as the tides slosh round and the air gets whipped up to super-hurricanes it does something really odd.
From a nice safe distance there's a bloke on another planet watching the earth. He sees me waving to him
As the earth turns I move across the surface of the earth (from his point of view) He looks at where I am and, by some odd coincidence he sees that my position traces out an exact sine wave (while I'm in sight) with respect to his local time.

On "big gun day" I happen to be in his field of view.
He sees me initially tracing out a sine wave. He sees me stop (I'm holding on really tight), then he sees me restart.
"Odd!" he thinks.
Then he goes back to looking at my traversing of the world and he spots that I'm now tracing out a cosine wave.
The cosine wave and the sine wave are both easy to do a FT analysis on. Each has exactly one frequency component, and it's the same.

Now, I may be mistaken, but I think that you can do the same thing with a smaller gun, if it only held the world back for 3 hrs the phase shift would be 45 degrees
The FT would give  equal sine and cosine components- but with amplitudes (If I have the maths right) of cos 45 and sine 45.
 If you only have a cow fart worth of phase shift, you only introduce a very small amount of cos theta into the equation- but what you don't do is add any other frequencies.