Guest Post by Willis Eschenbach
I’ve been listening to lots of stuff lately about tidal cycles. These exist, to be sure. However, they are fairly complex, and they only repeat (and even then only approximately) every 54 years 34 days. They also repeat (even more approximately) every 1/3 of that 54+ year cycle, which is 18 years 11 days 8 hours. This is called a “Saros cycle”. So folks talk about those cycles, and the 9 year half-Saros-cycle, and the like. The 54+ year cycle gets a lot of airtime, because people claim it is reflected in a sinusoidal approximately 54-year cycle in the for example the HadCRUT temperature records.
Now, I originally approached this tidal question from the other end. I used to run a shipyard in the Solomon Islands. The Government there was the only source of tide tables at the time, and they didn’t get around to printing them until late in the year, September or so. As a result, I had to make my own. The only thing I had for data was a printed version of the tide tables for the previous year.
What I found out then was that for any location, the tides can be calculated as a combination of “tidal constituents” of varying periods. As you might imagine, the strongest tidal constituents are half-daily, daily, monthly, and yearly. These represent the rotations of the earth, sun, and moon. There’s a list of the various tidal constituents here, none of which are longer than a year.
Figure 1. Total tidal force exerted on the Earth by the combination of the sun and the moon.
So what puzzled me even back then was, why are there no longer-period cycles used to predict the tides? Why don’t we use cycles of 18+ and 54.1 years to predict the tides?
Being a back to basics, start-from-the-start kind of guy, I reckoned that I’d just get the astronomical data, figure out the tidal force myself, and see what cycles it contains. It’s not all that complex, and the good folks at the Jet Propulsion Lab have done all the hard work with calculating the positions of the sun and moon. So off I went to JPL to get a couple hundred years data, and I calculated the tidal forces day by day. Figure 1 above shows a look at a section of my results:
These results were quite interesting to me, because they clearly show the two main influences (solar and lunar). Figure 1 also shows that the variations do not have a cycle of exactly a year—the high and low spots shift over time with respect to the years. Also, the maximum amplitude varies year to year.
For ease of calculation, I used geocentric (Earth centered) coordinates. I got the positions of the sun and moon for the same time each day from 1 January 2000 for the next 200 years, out to 1 Jan 2200. Then I calculated the tidal force for each of those days (math in the appendix). That gave me the result you see in Figure 1.
However, what I was interested in was the decomposition of the tidal force into its component cycles. In particular, I was looking for any 9 year, 18+ year, or 54.1 year cycles. So I did what you might expect. I did a Fourier analysis of the tidal cycles. Figure 2 shows those results at increasingly longer scales from top to bottom.
Figure 2. Fourier analysis of the tidal forces acting on the earth. Each succeeding graph shows a longer time period. Note the increasing scale.
The top panel shows the short-term components. These are strongest at one day, and at 29.5 days, with side peaks near the 29.5 day lunar cycle, and with weaker half-month cycles as well.
The second panel shows cycles out to 18 months. Note that the new Y-axis scale is eight times the old scale, to show the much smaller annual cycles. There are 12 month and 13.5 month cycles visible in the data, along with much smaller half-cycles (6 months and 6.75 months). You can see the difference in the scales by comparing the half-month (15 day) cycles in the top two panels.
The third panel shows cycles out to 20 years, to investigate the question of the 9 and 18+ year cycles … no joy, although there is the tiniest of cycles at about 8.75 years. Again, I’ve increased the scale, this time by 5X. You can visualize the difference by comparing the half-year (6-7 month) cycles in the second and third panels. At this scale, any 9 or 18+ year cycles would be very visible … bad news. There are no such cycles in decomposition of the data.
Finally, the fourth panel is the longest, to look for the 54 year cycle. Again, there is no such underlying sine-wave cycle.
Now, those last two panels were a surprise to me. Why are we not finding any 9, 18+, or 54 year cycle in the Fourier transform? Well … what I realized after considering this for a while is that there is not a slow sine wave fifty-four years in length in the data. Instead, the 54 years is just the length of time that goes by before a long, complex superposition of sine waves approximately repeats itself.
And the same thing is true about the 18-year Saros cycle. It’s not a gradual nine-year increase and subsequent nine-year decrease in the tidal force, as I had imagined it. Instead, it’s just the (approximate) repeat period of a complex waveform.
As a result, I fear that the common idea that the apparent ~60 year cycle in the HadCRUT temperatures is related to the 54-year tidal cycles simply isn’t true … because that 54 year repeating cycle is not a sine wave. Instead, looks like this:
Figure 3. The 54 year 34 day repetitive tidal cycle. This is the average of the 54-year 34-day cycles over the 200 years of data 2000-2200.
Now, as you can see, that is hardly the nice sine wave that folks would like to think modulates the HadCRUT4 temperatures …
This exemplifies a huge problem that I see happening. People say “OK, there’s an 18+ year Saros cycle, so I can divide that by 2. Then I’ll figure the beat frequency of that 9+ year cycle with the 8.55 year cycle of the precession of the lunar apsides, and then apply that to the temperature data …”
I’m sure that you can see the problems with that approach. You can’t take the Saros cycle, or the 54+ year cycle, and cut it in half and get a beat frequency against something else, because it’s not a sine wave, as people think.
Look, folks, with all the planets and moons up there, we can find literally hundreds and hundreds of varying length astronomical cycles. But the reality, as we see above, is not as simple as just grabbing frequencies that fit our theory, or making a beat frequency from two astronomical cycles.
So let me suggest that people who want to use astronomical cycles do what I did—plot out the real-life, actual cycle that you’re talking about. Don’t just grab the period of a couple of cycles, take the beat frequency, and call it good …
For example, if you want to claim that the combined tidal forces of Jupiter and Saturn on the sun have an effect on the climate, you can’t just grab the periods and fit the phase and amplitude to the HadCRUT data. Instead, you need to do the hard lifting, calculate the actual Jupiter-Saturn tidal forces on the sun, and see if it still makes sense.
Best regards to everyone, it’s still raining here. Last week, people were claiming that the existence of the California drought “proved” that global warming was real … this week, to hear them talk, the existence of the California floods proves the same thing.
In other words … buckle down, it’s gonna be a long fight for climate sanity, Godot’s not likely to show up for a while …
w.
THE USUAL: If you disagree with something that I or someone else said, please quote the exact words you disagree with, and tell us why. That way, we can all understand what you object to, and the exact nature of your objection.
CALCULATIONS: For ease of calculations, I downloaded the data for the sun and moon in the form of cartesian geocentric (Earth-centered) coordinates. This gave me the x, y, and z values for the moon and sun at each instant. I then calculated the distances as the square root of the sum of the squares of the xyz coordinates. The cosine of the angle between them at any instant is
(sun_x * moon_x + sun_y * moon_y + sun_z * moon_z) / (sun_distance * moon_distance)
and the combined tidal force is then
sqrt( sun_force^2 + moon_force^2 + 2* sun_force * moon_force * cos(angle))
DATA AND CODE: The original sun and moon data from JPL are here (moon) and here (sun), 20 Mb text files. The relevant data from those two files, in the form of a 13 Mb R “save()” file, is here and the R code is here.
EQUATIONS: The tidal force is equal to 2 * G * m1 * m2 * r / d^3, where G is the gravitational constant, m1 and m2 are the masses of the two objects, d is the distance between them, and r is the radius of the object where we’re calculating the tides (assuming that r is much, much smaller than d).
A good derivation of the equation for tidal force is given here.
charles nelson says:
February 9, 2014 at 10:45 pm
Are you really paying that little attention? I didn’t have to deal with any of that, not one bit.
Why not?
Because the Jet Propulsion Laboratory did all that for me. All of the relevant calculations, including any of those considerations you listed which might be relevant in my situation, are taken into account by JPL in their online ephemeris, which I used.
Read the head post before uncapping your electronic pen, charles, you’ll go further … and get laughed at less.
w.
David Falkner says:
February 9, 2014 at 9:30 pm
Um … we know this because astronomers check the position of the earth against the stars. A sidereal year is a bit more than 365 days. So if we use a 365 day year, and we start out with Aries in the spring, after 100 years will have drifted around the zodiac by 25 days or so, and if we don’t fix it pretty soon the spring equinox will be coming in the wrong season … so we adjust it.
See also Nylo’s comment above.
w.
Steven Mosher says:
February 9, 2014 at 10:28 pm
Scafetta … no surprise. He’s done immense damage to the skeptical cause. No shoes, no shirt, no service … no data, no code, no science.
I do note the irony, however, that to my knowledge the New Scientist has never busted a global warming activist for not showing data and code … but they’re happy to bust Scafetta. Ah, well, all publicity for revealing data and code is good publicity …
w.
I think the laughter is mostly directed at you Willis…and by the way do let us know when you ‘calculate’ how many angels can dance on the head of a pin.
[snip – ok, your off-topic rantings about the “iron curtain” have gone over to the loony zone now. I’m assigning you to the troll bin then since you’ve been warned previously – Anthony]
Hi Willis,
Thank you for your explanation, I have but one problem, some large lakes north of the moons track. Have small tides that are measurable. North of the moons path. The tide runs away from the moon, like the water is repulsed, this tends to bother my brain some what.
Maybe you have an explanation, if you have I would like to hear it.
JP says: February 9, 2014 at 10:32 pm
This is getting too much of a personal Eschenbach outlet here. Nothing personal, but I’m moving over to Bishop Hill for my climate news.
Ha ha .. that sort of statement is always cute! Can’t control your eyeballs, eh?
What I do if I don’t like something is to skip over it … You could skip every second article here and you’d still get more reading than at Bishop Hill (much as I do appreciate BH).
Hang on a sec, think of the free time I’d generate! Maybe I should follow your lead?
charles nelson says:
February 10, 2014 at 12:15 am
47, everyone knows that.
w.
wayne Job says:
February 10, 2014 at 12:32 am
Sorry, no clue … do you have more information, like an article on the subject or the like?
w.
Toto says:
This metaphor will only confuse. Tides are not sloshing in the open ocean, they are a wave, a very fast wave.
The problem is that people think that the molecules that propagate the wave are moving a lot. Hence they get the impression that there is a huge shift in the water with each tide, that will somehow mix it, and mix it more if the tides are a bit bigger.
Whereas, a wave doesn’t require very much movement at all by its individual components. They just have to align to move in the same direction for a brief period of time.
The idea that anyone would seriously consider that the long run cycles in tides affects the amount of mixing bothers me.
(I note that many people seem to think that winds are somehow affecting the amount of mixing. I find that very difficult to believe. The surface layer isn’t going to be still at the best of times, so more mixing of a well mixed top couple of metres makes little difference. I doubt strongly that winds cause deep mixing beyond that layer. The concept that winds can cause water to not just pile up, but then force it downwards seems very unlikely IMO. The overwhelming power of the Humbolt Current, Gulf Stream etc are orders of magnitude more important.)
Willis, I do disagree, but its hard to cite what, but I do think your conclusion is unjustified. Recently at a conference I was treated to a very interesting lecture about how the great floods of Queensland, Australia are correlated to solar and lunar juxtaposition. His premise is that floods occur when the sun and moon are both at apogee in the wet season, since the orbital patterns repeat each 18 years the weather effect has an 18 year cycle (except I guess it is the earth at apogee, but you knew what I meant right?). Interestingly this occured in both summer of 2011 and a less pronounced peak in summer of 2013, both of which had great floods.
Anyway, while the tidal forces themselves may have no pattern the way they interact with the seasons and the monsoon certainly does show a pattern. At least in northern Australia.
Where, and when the tidal forces peak is important.
The eccentricity of the moon’s orbit around the earth is not constant and is varying on a monthly to yearly basis. I know this sounds crazy – but it really is true. If you plot out the eccentricity values of the moon relative to the earth-moon barycentre using JPL ephemeris – for example take their horizon interface you will get the following :
graph shown here
There are at least 2 regular resonances which at first sight seems odd because neither coincide with the orbital period of the moon (27.32days) nor that of the earth (365.25 days). There are also beats in the amplitude. Following this german article, I made a least squares fit shown as the blue curve which reproduces almost perfectly the signal .
eccentricity(d) = 0.55 + 0.014cos(0.198*d + 2.148) + 0.0085cos(0.0305*d +10.565)
This variation in eccentricity changes the perihelion distance from the earth significantly causing large variations in the strength of spring tides on a yearly basis. The eccentricity becomes a maximum when the semi-major axis of the orbit lines up with the sun. This happens every 205.9 days – more than half a year due to the precession of the orbit every 18.6 years. The 31.8 day variation is I think the regular orbital change in distance from the sun.
The moon is really in orbit around the sun because the sun’s gravitational field on the moon is twice that of the earth’s. The moon’s orbit is locked into that of the earth to give an effective lunar orbit as viewed from earth. It turns out to be impossible to accurately calculate the moon’s effective orbit around the earth far into the past. The error on the lunar eccentricity becomes > 100% more than 1 million years ago as reported in Laskar et al. 2010. Who knows what happens to the lunar eccentricity when the earth’s eccentricity around the sun increases with Milankovitch cycles ? Large changes in the lunar-earth distance will have very large (1/R^3) effects on tides and indirectly on climate.
“Thank you for your explanation, I have but one problem, some large lakes north of the moons track. Have small tides that are measurable. North of the moons path. The tide runs away from the moon, like the water is repulsed, this tends to bother my brain some what”.
It’s because teh tidal works in both directions ! Yeah, sounds mad and is a bit hard to imagine but the tidal force is due the gradient or divergence of the gravitational field.
Graivity falls off as inv sqr law : 1/r^2 the rate of change with respect to radial distance is thus proportinal to 1/r^3 , hence the r^3 in the formula.
So it’s not just a simple gravitational tug as one would think intuitively.
So the nearest part of the ocean gets more gravitational attraction than the centre of the Earth, Equally the ocean at the opposite side to the moon gets _less_ attraction in about the same measure when compared to the centre of the Earth. The Earth gets accelerated towards the moon more strongly than the far ocean which gets ‘left behind’ so to speak.
That is why many tides tend to happen twice per day. That’s called semi-diurnal. This is a simple case when the moon is over the equator.
When the moon is not over the equator one of these simplistic “bulges” (which don’t actually happen like that in reality) is circling to the north and the other to the south. Thus it is a once a day event as the Earth rotates.
There is some overlap and the result a composite tide with both diurnal and semi-diurnal components.
It took me a long time to find that out because there is an enormous amount of misunderstanding and misinformation even from academic sources.
Hopefully this will help others understand with a lot less effort.
Willis Eschenbach says:
February 9, 2014 at 5:33 pm
“The key word being “tiny” … look, Richard, any planet, moon, asteroid, and planitesimal affects the orbit every single other planet, moon, asteroid, and planitesimal. That’s not the question.”
Ok, this is going to get long. Sorry in advance for those who have to wade through it.
So we have Earth, Moon and Sun in a constant, never quite repeating gravitation dance with very minute additional factors that influence both of the TWO components of the gravitational field and its impact here on Earth.
Now I really do not know the answer to all this and it may well end up in the ‘Meh – Who cares’ bucket that you (and Leif) so quickly placed it in but, as people are calculating temperatures to 1/100s of a degree C, I hope you will bare with me whilst I try to lay the case out.
Firstly the vertical component to the field is as you plotted at the top. This should strictly be plotted around the barycentre (but others have mentioned that already). The vertical field is every so egg shape – pointy end towards the Moon – as this is the sum of two different forces to make the one field (otherwise we would have only one tide a day). Picky points but 1/100s of a degree is picky also. Now this plot gives the vertical field on a water only globe (rotating or not as you wish).
We still have the other component, the tangential to the surface field that happens at 45 to 60 degrees to the orbital plane as shown by me above (from Wiki). This is the part of the picture, overlooked so often, that I believe may be what is the important bit here. This is a force not seeking to raise the water/atmosphere up and down but to push it sideways back and forth along the surface. It operates across the whole cross-section of the water/air column and does not have the same high speed pattern that the vertical vector does because it is an almost constant angle despite the rotations.
So we have the two ellipses of the two orbits beating horizontally now. We are still missing the other major factor. The vertical beating. The two orbits are not aligned to each other vertically. So there is another supple beat happening there as well. That IS influenced by Saturn in particular which moves the path up and down (I know not very much but details again).
Whilst we are at it lets bring in the fact that the Earth is not rotating at right angles to the orbital plane either. Not important so much if this was a water only globe but if that was the case we wouldn’t be here to worry about this stuff.
With a land/ocean mixed globe this now becomes much more important. Now we have to worry about fixed points on the globe as it rotates. That brings in the Leap Year ‘same point in the sky, same time of day, same day of the year’ detail as well as the Saros cycle.
Now we have the ‘lever’ reasonably well defined and it modulates around its central value but how does that operate here on Earth. What ‘fulcrums’ are there against which it can operate?
Now land/ocean geography plays its part.
That tangential field. It operates at 45-60 degrees or so, and the important Geography appears to lie in the Northern Hemisphere. And the vertical field (much smaller) also operates on the North Pole (and the South Pole as well but that is all land). This vertical component operates with the same periodicity as daylight and we all know how daylight operates at the poles. Not the same quick pattern as tides/daylight elsewhere.
So we have a multi chamber, geographically defined, pumping system driven by the vertical field and ported to the rest of the rest of the Oceans through some very small Straights and Cills that surround the Northern Oceans. This sucks water in and out through the Straights and the resultant opposes or helps the currents that otherwise flow through them.
Then we have the tangential component also helping or preventing current flow though the other major Straights and Cills between Greenland to Scotland.
These are tiny flows of a few knots at best and it is the percentage change that matters, not the absolute value. Anyone who sails knows that tidal stream can well affect motion over the land and this is the sort of effect we are seeking. A few meters rise in the Ocean surface if a Cill height is only a few hundred meters or less matters also. Current will flow much easier in one case than the other.
The atmosphere may also be of interest when considering the tangential component as well. This 45-60 degree band is precisely where the Polar and Ferrel Cells meet. Does that ‘along the surface or slightly upwards/downwards’ vector affect where it forms or not?
And now we are in territory that is well beyond that of simply downloading data from JPL and running a plot. This is super-computer land and I don’t have one of those to hand.
Hence the curiosity.
Can simple, physics based, field vector maths and some fluid dynamics explain the longer term patterns we see? No exotic theory. Just a simple application of known physics to the case in hand.
Anyone got a research grant? Oil money?……
Willis Eschenbach says:
February 10, 2014 at 1:08 am
“47, everyone knows that.”
I thought it was 42 (and that mice and dolphins were the reason) 🙂
Venus and Earth trajectories around Sun can be thought of as independent periodical signals. And you may say that there’s nothing like a 243-year period after which the cycle repeats, it’s just how the two independent signals add up.
Except that Venus transits over Sun happen at such period.
I don’t really disagree with any of your conclusions, but I can see your methods. When you want to prove there is a period, you use periodicity analysis and declare it better than fourier transform. When you want to prove there is not the period, you use fourier transform.
The truth is, the strongest tidal effect event happens with the 54-year period. It may not be much stronger than other local maxima, but similarly to the Venus transits, there is certain period to them.
Clive Best: “The eccentricity of the moon’s orbit around the earth is not constant and is varying on a monthly to yearly basis. I know this sounds crazy – but it really is true. ”
It’s not that odd. The changes in the moon’s orbit are due to the graviational effect or the other planets. How they interact and line up will be very complex. Also the sun is not stationaly with respect to a inertial frame of reference. It has its own path relative to the solar system barycentre.
Since the earth and the moon are orbitting the sun we get pulled around with it.
So Willis is quite correct that the direct effect of planetary gravity on Earth is neglibible but it is wrong to conclude that the postition of the planets has no effect or the Earth or the moon.
The moon’s orbit is somewhat eccentric, at it’s most extreme the difference between closest approach (perigee) and furthest distance (apogee) is about 14% . The produces a difference of around 40% in r^3 hence the lunar tidal force. It goes from one extreme to the other and back in 27.55 days.
I have detected this period in the Arctic ice data:
http://climategrog.wordpress.com/?attachment_id=757
The alignment of the this excentricity is called the line of apsides and it too rotates with respect to rest of the system with a period of 8.85 years
It really requires some mental gymnastics to try to visualise all this but it is not neither simple nor negligible, in view of the 40% change in tidal force.
Oh, and just for added fun that 40% goes up and down about twice a year ( in fact rather more than 6 months).
I think that’s what Clive’s plot was showing.
Willis
The article seems rather terse. Could you answer a few queries:
1:
I’ve been listening to lots of stuff lately about tidal cycles. … So folks talk about those cycles, and the 9 year half-Saros-cycle, and the like. The 54+ year cycle gets a lot of airtime, because people claim it is reflected in a sinusoidal approximately 54-year cycle in the for example the HadCRUT temperature records.
There is no background literature provided, you have the reader at a disadvantage. Is there any that pertains to this data/approach.
2:
Your Fourier analysis plot is a bit unconventional (by my experience). Are your amplitudes computed as the magnitude of the complex form (sin and cos). It’s seems an unusual way to plot these; mixing frequency and time domain. I’m guessing you have simply converted wave numbers to their wavelength for ease of expression?
Have you looked at the phase spectra? Would this offer anything?
3:
I don’t understand your equation:
sqrt( sun_force^2 + moon_force^2 + 2* sun_force * moon_force * cos(angle))
I don’t get this. Surely you should just modulate the sun_force with the magnitude of the dot product (i.e. |cos(angle)|):
sun_force = sun_force * |cos(angle)|. Then:
F=sqrt(sun_force^2+moon_force^2);
Perhaps not but could you explain why?
4:
Finally, what is the point of the article? What is the take home message? And why is it important?
Is it that the 54 cycle point is lost do you mean there is meant to be one and you’ve showed there isn’t. I could be missing something but you don’t seemed to have shown this.
Mosh’ says: “that’s a man of his convictions.. Not.”
No that’s a man who is prepared to make a correction when he overstated the case in a published article.
Constast that to someone like lsvalgaard Vaghan Pratt who will argue till they’re blue in the face rather than admit he made a mistake. That kind of “conviction” I can do without.
Scafetta’s correction seems honorable. At least he has the humility to admit he overstated the case and correct it. That should be applauded, not used in a mud slinging exercise.
If you try to shoot down people every time they correct themselves you lessen likelihood of it happening.
That is just yahboo politics, not scientific debate.
Willis , are you able to provide a precise value for the peaks in fig2 ?
In particular the ones that look to be circa 27, 29 days and 13 months.
Scafetta could satisfy 99.9% of his critics with a full release of data and code in order to enable replication of his papers, which currently cannot be treated as much more than anecdotes. Then he could be actually shown to be right or wrong and the majority will go where logic goes. As long as he continues to evade his responsibility as a scientist, criticism will increase, not abate.
charles the moderator says:
February 10, 2014 at 3:06 am
“Scafetta could satisfy 99.9% of his critics with a full release of data and code in order to enable replication of his papers, which currently cannot be treated as much more than anecdotes.”
Well I have continuously shown that there is a ~60 year pattern to the data, with or without Scafetta data and code.
http://i29.photobucket.com/albums/c274/richardlinsleyhood/Extendedtempseries-secondpass_zps089e4c7d.gif
Now with added proxy data as well to satisfy Jai who is SO convinced that it does not exist as well 🙂 ).
This is just simple ‘Gaussian’ low pass filter stuff but it independently confirms at least part of his case.
AW says: Well, that frequency certainly took a beating.
Willis:”At this scale, any 9 or 18+ year cycles would be very visible … bad news. There are no such cycles in decomposition of the data.”
Once thing you may not be realising Willis, and relates to all the comments I left yesterday trying to explain the way cycles combine in what we may loosely call “beats” or modulation.
The point is the equivalence of modulation (two signals multiplied together) and two different cycles that are added.
Now fourier analysis, by definition only captures single frequencies of fixed amplitude. So if you have data with a modulation it can not detect it as such. Instead what you get in the spectrum is two cosines that , added together, mathematically equivalent to the modulated form and will reproduce it perfectly if added together.
This is the old half-the-sum * half-the-difference game again. (Talking in freq, not period).
Again you can see an example in my arctic ice plot.
http://climategrog.wordpress.com/?attachment_id=757
There is 4.31 year modulation but this does not show up in the spectral analysis as 4.31 years. It shows as a triplet of peaks at
p1=27.1256
pc=27.6006
p2=28.0939
The central peak may sometimes be negligible . It is p1 and p2 that represent the modulation that in reality is 4.31 years. ie in a totally different part of the spectrum.
Similarly when you look at charts of tidal periods these are also “fourier” components and may (will) add together to give long period cycles. Thus absence of long periods in the list does not mean there are not long period cyclic patterns in tides.
There is clearly some kind of modulation pattern in your figure 3
It looks pretty sinusoidal and constant across that data, so probably results from just two components.
Greg: Lief: Willis: Whoever…
This thing about adding or multiplying for frequencies. The true answer is that it is always both.
So we get FM radio type stuff with addition where the frequencies are a long way apart.
And we get moiré interference patterns where the two are closer together.
For frequencies that are neither obviously one or the other often both factors are visible quite easily.
Thus 60 and 4 gives 4, 56, 60, 64 and 240.
2000 and 4 gives 4, 1996, 2000, 2004 and 8000.
N. Scaffeta also used ephemeris data in one of his papers to look at how the moon affected Earth’s orbit of the sun. By comparing spectra of Earth speed relative to the sun and that of the Earth-Moon ensemble (speed of EM barycentre) he showed that there was a 9.1 year variation that was caused by the presence of the moon.
I thought that was pretty ingenious.
Richard, “So we get FM radio type stuff ”
frequency modulation is another can of worms entirely , is that what you meant?