The period of a composite wave

In summary, the sum of multiple functions with different periods will have a period that is the same as the longest of the periods of the individual functions.
  • #1
roam
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TL;DR Summary
I am trying to understand whether the period of a composite wave relates to the relative phases or the number of the constituent waves.
I am working with a simulation which generates multiple identical functions that overlap differently (i.e., they are generated with randomly different phases from each other).

When I calculate the composite wave, the shape of the combined wave will differ depending on the relative phases of the input waves. But regardless of how the waves overlap, the "effective period" of the combined wave seems to remain the same. Why is that? What would be an analytic expression that explains this phenomenon?

For instance, this is the simulation using three waves (the thicker line is their sum):

242406


In addition, if we increase the number of waves, the temporal period of the combined wave will still remain constant. Why?

Here is an example:

242407


Any explanations would be appreciated.
 
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  • #2
I don't understand what these "waves" are. It seems to be simply a peaked function that is repeated periodically.

Also, the thicker line doesn't appear to be the sum of the other curves, but rather the average value of the sum.

roam said:
When I calculate the composite wave, the shape of the combined wave will differ depending on the relative phases of the input waves. But regardless of how the waves overlap, the "effective period" of the combined wave seems to remain the same. Why is that? What would be an analytic expression that explains this phenomenon?
If the underlying phenomena have the same period, then any linear combination of them will have the same period.
 
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  • #3
Hi DrClaude,

DrClaude said:
I don't understand what these "waves" are. It seems to be simply a peaked function that is repeated periodically.

Also, the thicker line doesn't appear to be the sum of the other curves, but rather the average value of the sum.

The function I am using is the complementary Airy function that gives the ratio of light reflected from an interferometer.

The thick line is the normalized sum of the individual curves.

DrClaude said:
If the underlying phenomena have the same period, then any linear combination of them will have the same period.

Do you know of an analytic expressions that shows that the combination will always have the same period (regardless of the phases or the number of the individual constituents)?
 
  • #4
Consider a set of functions ##\{ f_i(x) \}## of period ##a##, i.e., ##f_i(x+a) = f_i(x)##. Taking ##F(x)## as a linear combination of such functions,
$$
F(x) = \sum_{i=1}^{n} c_i f_i(x)
$$
then
$$
F(x+a) = \sum_{i=1}^{n} c_i f_i(x+a) = \sum_{i=1}^{n} c_i f_i(x) = F(x)
$$
therefore ##F(x)## is periodic with period ##a##.
 
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  • #5
I would say that the 'period' of a periodic function is (strictly) the interval between points when the waveform repeats. If the waveform consists of a number of functions, each of which has its own identifiable period (sine waves or strings of pulses, for instance), then the period of the composite function would be the lowest common multiple of the individual functions. This could be a very long period if the periods have, say periods of large prime numbers of seconds. This would, for instance, correspond to the 'beat' between two sine waves.
 
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  • #6
sophiecentaur said:
This could be a very long period if the periods have, say periods of large prime numbers of seconds.
If the ratio of the periods of the individual functions is an irrational number, their sum may not be periodic at all.
 
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  • #7
jbriggs444 said:
If the ratio of the periods of the individual functions is an irrational number, their sum may not be periodic at all.

This is an interesting point. So, if there are more than two individual signals present, then the ratio of the periods of each pair of signals must be rational. Otherwise, the signal will not be periodic. Is that correct?

Also, do you think this happens often in real life where the two signals have incommensurable periods?
 
  • #8
jbriggs444 said:
If the ratio of the periods of the individual functions is an irrational number, their sum may not be periodic at all.
I don’t think the term “irrational” is applicable here.
“Irrational” measurements don’t occur. All you can expect is a long recurrence in the ratio of the periods. (Longer than any possible experiment could last.)
 
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  • #9
sophiecentaur said:
I don’t think the term “irrational” is applicable here.
“Irrational” measurements don’t occur. All you can expect is a long recurrence in the ratio of the periods. (Longer than any possible experiment could last.)
Oh, come on! Who said anything about measurements? This is about the sum of mathematical functions.
 
  • #10
DrClaude said:
Oh, come on! Who said anything about measurements? This is about the sum of mathematical functions.
I have to agree. But the OP is discussing numerical simulation, which is where I got the idea of practicality.
 
  • #11
I am simulating a physical situation where optical fibers can host more than one spatial modes. From what I understood, if you don't experimentally see a period to such a signal on an oscilloscope, it could be the case that the ratio of the period of the constituent modes is not rational (or rather it has a longer recurrence than the length of the experiment).
 
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  • #12
roam said:
not rational (or rather it has a longer recurrence than the length of the experiment)
Bearing in mind that you can't produce irrational numbers in a numerical simulation, we should settle for 'a very long repeat time', I think.
 
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  • #13
roam said:
the ratio of the periods of each pair of signals must be rational. Otherwise, the signal will not be periodic. Is that correct?
For all practical purposes, yes, that would be correct.

Mathematically, there is a loophole. Suppose that you have two signals with period x that sum to zero and a third signal with period kx for some irrational number k. Then the sum of all three signals will be periodic despite two component signals having periods with an irrational ratio.
 
  • #14
I think we’re in the fuzzy, oxymoron region here - between axiomatic arguments and practical experience. I don’t Think the thread can reveal much more.
 
  • #15
I'm still trying yo figure out how a ratio could ever be 'irrational.'
 
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  • #16
Dullard said:
I'm still trying yo figure out how a ratio could ever be 'irrational.'
What about ##\sqrt{2}/2##? :smile:
 
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  • #17
OK. I should have said: A ratio of periods. Can the period of a waveform be irrational?
 
  • #19
DrClaude said:
What about ##\sqrt{2}/2##? :smile:
Point!
 
  • #20
DrClaude said:
What about ##\sqrt{2}/2##? :smile:
In the case of a one dimensional quantity like frequency, is there a way that an irrational frequency can occur? You can get a line that is 'exactly' √2 inches long but only in a two dimensional figure.
Also, I was thinking about transcendental numbers like e and π. Apart from passing through values involving, say e, can you get a steady frequency of e X 106 Hz? This is a question for someone well versed in Analysis, I think.
 
  • #21
sophiecentaur said:
In the case of a one dimensional quantity like frequency, is there a way that an irrational frequency can occur?
If we grant the assumption that frequencies are precisely knowable then the natural assumption would be that irrational frequencies are the norm and that rational frequencies are the exception. Viewed on a one-dimensional dartboard of real numbers, the rational numbers are a target set of measure zero.

But let us make it practical. If I have a piano string with a density of 2 grams per centimeter and a second piano string with a density of 1 gram per centimeter with both strings the same length and tension then the ratio of the frequencies should be ##\frac{\sqrt{2}}2##

Assuming I am not being a bonehead interpreting the string frequency formula.
 
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  • #22
roam said:
I am simulating a physical situation where optical fibers can host more than one spatial modes. From what I understood, if you don't experimentally see a period to such a signal on an oscilloscope, it could be the case that the ratio of the period of the constituent modes is not rational (or rather it has a longer recurrence than the length of the experiment).
Okay, so optical frequencies are like ##10^{14}## hertz or more. My scope cuts out around ##10^8## Hertz. Can I ask where you purchased yours?
 
  • #23
jbriggs444 said:
If we grant the assumption that frequencies are precisely knowable then the natural assumption would be that irrational frequencies are the norm and that rational frequencies are the exception.

But let us make it practical. If I have a piano string with a density of 2 grams per centimeter and a second piano string with a density of 1 gram per centimeter with both strings the same length and tension then the ratio of the frequencies should be ##\frac{\sqrt{2}}2##

Assuming I am not being a bonehead interpreting the string frequency formula.
I can see where you're coming from and the Maths makes you right but you are talking in purely theoretical terms.
I seem to remember that , from Archimedes, it can be shown that between any two rational numbers there will always be an irrational number which lies in between them. So with your Maths hat on, you have to be right. This implies that signals of any two frequencies, generated totally independently, would have an indeterminately long repeat period.
Frequencies, generated from the same frequency reference, will have rational values (on that reference scale), though.
 
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  • #24
sophiecentaur said:
it can be shown that between any two rational numbers there will always be an irrational number which lies in between them
Yes. And vice versa. Both the rationals and the irrationals are dense in the reals. No matter where you land a dart, there will be both irrational and rational numbers arbitrarily close.

From a practical point of view, this means that in any experiment with any margin of error, no matter how small, the measured result will be consistent with infinitely many true values that are rational and infinitely many true values that are irrational. [Making the notion of a "true value" rather questionable].

But that takes all the fun out of the game. So I'm putting my funny looking math hat back on.
 
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  • #25
jbriggs444 said:
So I'm putting my funny looking math hat back on.
I know a number of Maths teachers and worked with a number of Mathematicians. They were nice people but slightly (in some cases, even more than slightly) nutty. I think the subject has a way of getting its own back on Mathematicians. How can you be rational when at least half of the stuff you are dealing with is irrational?
 
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  • #26
In many ways, periodic functions are a fiction, a useful mathematical abstraction used to describe nature under limited circumstance. Since the universe is finite in time, there are no truly periodic functions.
 
  • #27
roam said:
But regardless of how the waves overlap, the "effective period" of the combined wave seems to remain the same. Why is that?

The good doctor has already explained it, but anyway, to the contrary, you should expect the effective period to remain the same, and in fact, it would be very surprising if it didn't.
 
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  • #28
I don't know about anyone else, but the question raised in #2 wasn't addressed AFAIK. A reflection coefficient is not a wave. What's plotted in #1 appears to be the magnitude of the reflection (transmission?) coefficient for X where X is what exactly?
 
  • #29
Paul Colby said:
I don't know about anyone else, but the question raised in #2 wasn't addressed AFAIK. A reflection coefficient is not a wave. What's plotted in #1 appears to be the magnitude of the reflection (transmission?) coefficient for X where X is what exactly?
You're right. If there's reflection then the time and spatial variations come into it. Or is it just the repeating interference pattern due to two signal paths that's being discussed? I don't see how any such diffraction patterns will ever repeat over a wide angle.
 
  • #30
Yeah, I agree. The reason to ask is this impacts what operations are done on said "signal". Looks like maybe a scanning fiber interferometer of some sort where (I'm guessing) the "waves" are fiber modes. A great deal more need be supplied IMO.
 
  • #31
Paul Colby said:
Okay, so optical frequencies are like ##10^{14}## hertz or more. My scope cuts out around ##10^8## Hertz. Can I ask where you purchased yours?

I am simulating fiber ring resonators which basically behave as Fabry-Perot interferometers. What I am plotting is the output of the ring given by the Airy function. The frequency spacing between the adjacent resonance dips (i.e. the period, also known as the free spectral range) is typically ~1-2 MHz for a 100 m loop. When you have multiple modes, they will each have a slightly different period.
 
  • #32
roam said:
When you have multiple modes, they will each have a slightly different period.
This seems to be a general principle in Physics; it corresponds with the fact that the Overtones in many resonant systems are not, in fact, harmonics due to 'end effects' (whatever that means in some cases)
 
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  • #33
Dullard said:
I'm still trying yo figure out how a ratio could ever be 'irrational.'
A ratio being an irrational number may look contradictory at first pass, but rational numbers are those which can be expressed as ratios of integers, and not all ratios can be so expressed; real numbers which can't be so expressed are called irrational, even if they are real ratios, e.g., the ratio of the circumference of a circle to its diameter is an irrational number: ##\frac c d =\pi##.
 
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  • #34
sysprog said:
A ratio being an irrational number may look contradictory at first pass, but rational numbers are those which can be expressed as ratios of integers, and not all ratios can be so expressed; real numbers which can't be so expressed are called irrational, even if they are real ratios, e.g., the ratio of the circumference of a circle to its diameter is an irrational number: ##\frac c d =\pi##.
I think the problem we are having here is that we are told by Mathematicians about the set of real numbers but we do not actually 'see' quantities that are not rational. Yes, the diagonal of a 1cm square is irrational but we could never cut a piece of wood that's √2 cm long. Our minds work in a quantised world (except when we step into the Maths Jungle) and, of course, any quantised value is, in the end, rational.
This sort of problem goes way back and 'squaring the circle' has bothered people for hundreds of years.
 
  • #35
roam said:
I am simulating fiber ring resonators which basically behave as Fabry-Perot interferometers.
Thanks, this should have been the sentence to start the thread. The degree to which one may abstract your question depends on what you actually mean by "signal" "wave form" even function is too vague in this case IMO.

So, at the heart of your simulation is a truly linear system, time harmonic or band limited EM waves in a fiber. Your signal is a time averaged normalized intensity (absolute square) of the interferometer output as a function of interferometer length. The scan frequency is not relevant, it's just a means of reading out the data.

We (well, I actually) can only guess as to what this simulation contains in the way of assumptions or details. Just in general terms the guide or fiber wavenumber will depend on the mode in a complex way depending on nearly everything. There is no reason to expect the ratio of these mode wave numbers to be integer or rational values.
 
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<h2>1. What is a composite wave?</h2><p>A composite wave is a type of wave that is made up of two or more individual waves with different frequencies, amplitudes, and wavelengths. These waves can overlap and create a complex pattern.</p><h2>2. How is the period of a composite wave calculated?</h2><p>The period of a composite wave is calculated by finding the lowest common multiple (LCM) of the individual wave periods. This can be done by dividing the individual wave frequencies into the LCM.</p><h2>3. Can the period of a composite wave be longer than the individual wave periods?</h2><p>Yes, the period of a composite wave can be longer than the individual wave periods. This is because the waves can overlap and create a more complex pattern, resulting in a longer period.</p><h2>4. How does the amplitude of individual waves affect the period of a composite wave?</h2><p>The amplitude of individual waves does not directly affect the period of a composite wave. However, it can affect the overall shape and intensity of the composite wave.</p><h2>5. What are some real-life examples of composite waves?</h2><p>Some real-life examples of composite waves include sound waves, which are made up of multiple frequencies, and ocean waves, which can be a combination of wind waves and swells from distant storms.</p>

1. What is a composite wave?

A composite wave is a type of wave that is made up of two or more individual waves with different frequencies, amplitudes, and wavelengths. These waves can overlap and create a complex pattern.

2. How is the period of a composite wave calculated?

The period of a composite wave is calculated by finding the lowest common multiple (LCM) of the individual wave periods. This can be done by dividing the individual wave frequencies into the LCM.

3. Can the period of a composite wave be longer than the individual wave periods?

Yes, the period of a composite wave can be longer than the individual wave periods. This is because the waves can overlap and create a more complex pattern, resulting in a longer period.

4. How does the amplitude of individual waves affect the period of a composite wave?

The amplitude of individual waves does not directly affect the period of a composite wave. However, it can affect the overall shape and intensity of the composite wave.

5. What are some real-life examples of composite waves?

Some real-life examples of composite waves include sound waves, which are made up of multiple frequencies, and ocean waves, which can be a combination of wind waves and swells from distant storms.

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