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The frequency of rotation was always 1 in 24 hrs. What the cows did was slightly affect the phase.
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)
but it would constitute a real frequency shift rather than just a phase shift.
A single one-off event in the whole of time doesn't produce a frequency shift.
Since the frequency spectrum is the Fourier transform of the signal, terms like "frequency change" are somewhat problematic.
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.
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.
For your case, I believe it was 2*pi radians and you started right when the cows farted?
Quote from: JP on 02/08/2011 13:52:08For your case, I believe it was 2*pi radians and you started right when the cows farted?That would work.
It doesn't repeat.It has no repetition.Since it happens once, and never again, it does not happen more than once.
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.
Geezer,you say "During IFD the angular velocity was not uniform. It increased a bit, then it slowed back down so that the daily cycle time was reduced and that resulted in a phase shift of the Earth's rotational cycle relative to our atomic clock "day" (pretty hard to argue with that)."Yes, the earth had an off day in terms of timekeeping.But only one bad day in the whole of forever. On average, it didn't happen.
Did someone say that the Earth is a closed system? Does that exclude, Light, gravity, cosmic rays, dark matter, asteroids, meteors, comets and UFOs? Not to mention interplanetary dust. In the cosmic context I do not think earth is a closed system, if it was we could not exist as the dinosaurs would still be running about.. 
Geezer, I'm averaging over an infinite past history. (which is one such simplifying assumption).
If it only happens once, you can chalk it up to observational error and ignore it, right?
Ok, I just came up with this argument that seems to prove by contradiction that "fart day" generates extra frequencies (via the Fourier transform). Let me know what you think.First, you need to know that the Fourier transform has an inverse. From a signal over time, you can uniquely get the frequency spectrum of that signal, and from a frequency spectrum you can uniquely recover the signal over time. Second, if the earth rotated unimpeded by farting cows, you could model it by a periodic sinusoid, s(t). Maybe this sinusoid starts and stops and maybe it goes off to infinity. It doesn't matter. It generates a frequency spectrum, say S(f). You can go back and forth from S(f) to s(t) by Fourier transforms and inverse Fourier transforms. I could just as easily have told you that the frequency spectrum is S(f) and you could have recovered the signal over time, s(t). There is no loss of information in the Fourier transform.Let's assume BC is right and that the frequency spectrum with the cows farting is the same as without. If that's the case, then it's also given by S(f). By the properties of the Fourier transform, a frequency spectrum S(f) means that the earth's rotation is given by s(t), which we know is true from above.But this is identical to the signal without the cows farting, and we know the signals cannot be identical (there's a phase shift). So the frequency spectra cannot be identical.QED?
To take a simple example the signal function (sin qt + sin 2qt) has precisely the same frequency spectrum as (sin qt + cos 2qt) -- equal spikes at f = q and f = 2q -- but they are quite different functions.
I do think it has problems representing an instantaneous phase change in a sine wave, for example, but this isn't physical.
since you can't sample the step with perfect resolution.
Quote from: JP on 05/08/2011 21:35:34since you can't sample the step with perfect resolution.Ah, right! Doesn't that boil down to giving it a certain amount of slope that doesn't really exist, in which case a step function would really be a ramp?
Something like that, I think. When you actually sample the function, you have to do a discrete Fourier transform, which isn't quite the same as the continuous integral. If the function has sharp features that you can't resolve perfectly with your sampling, it throws off the result a little bit.
Quote from: JP on 06/08/2011 12:43:59Something like that, I think. When you actually sample the function, you have to do a discrete Fourier transform, which isn't quite the same as the continuous integral. If the function has sharp features that you can't resolve perfectly with your sampling, it throws off the result a little bit.Right, but what about a "vertical" section of the signal? If it really is vertical, there is no phase angle between the start and end of the vertical section.
The way I'm picturing this, the signal is sin(t) to the left of the discontinuity and sin(t+phi) to the right. The phase of the sine shifts by phi at the discontinuity.
I'm pretty sure an instantaneous phase change constitutes a vertical "jump". If the signal had a horizontal section and you analysed only that section, there would be no frequency components at all, whereas the vertical jump tends towards infinite frequencies.EDIT: Of course, if you do the "jump" at precisely the right time, the signal would just have a sharp bend in it.
Great! Well, I think this cow has been more than adequately flogged.