Frequency Doubling: Interaction of 2 Waves Explained

[DariusP]

How can two waves interacting with each other produce a wave with double the frequency? I always thought two waves can only interfere constructively or destructively, hence, increasing or decreasing their amplitude but keeping frequency the same.

 

[osilmag]

Imagine a 230 Hz and 220hz wave in CONTINUOUS time. There will be some overlap between those two frequencies if you add them together. That is the extra frequency that is detected.

 

[CWatters]

Look at "Trig Identities" eg sin(X) + sin(Y) = ...

 

[sophiecentaur]

Imagine a 230 Hz and 220hz wave in CONTINUOUS time. There will be some overlap between those two frequencies if you add them together. That is the extra frequency that is detected.

There can be no "overlap" between two different single frequencies. You can only get overlap where two signals of finite bandwidth share part of the spectrum. Overlap doesn't constitute an explanation for frequency doubling, I'm afraid. You need an appropriate model for this.
If you could only believe the Maths of this, it would just 'fall out'. Arm wave can never be as convincing because the description is really not adequate.
If you take two sinusoidal signals, V₁ = A sin(ω₁t) and V₂=A sin(ω₂t) and subject them to a non linearity, say a simple square law where
V_out =V_in²
= (V₁ + V₂)²

Write that in full V_out = (A sin(ω₁t) + A sin(ω₂t))² then expand it and follow @CWatters idea of using trig identities.
You get:
V_out = A²((sin(ω₁t)² + (sin(ω₂t)²)+2sin(ω₁t sin(ω₂t))
The left hand two terms will contain 2ω₁ and 2ω₂ and the right hand term will contain terms with ω₁+ω₂ and ω₁-ω₂ which corresponds to second harmonics of the input signals and sum and difference frequencies.

 

[Charles Link]

In non-linear optical materials, there is a response of the material proportional to the second power of the electric field strength in the electromagnetic field. The result is a frequency doubling of a portion of the incident beam. See also https://en.wikipedia.org/wiki/Second-harmonic_generation

 

[DariusP]

In non-linear optical materials, there is a response of the material proportional to the second power of the electric field strength in the electromagnetic field. The result is a frequency doubling of a portion of the incident beam. See also https://en.wikipedia.org/wiki/Second-harmonic_generation

I know this and I don't get this

(if I didn't hear about this I wouldn't be able to ask)

 

[DariusP]

There can be no "overlap" between two different single frequencies. You can only get overlap where two signals of finite bandwidth share part of the spectrum. Overlap doesn't constitute an explanation for frequency doubling, I'm afraid. You need an appropriate model for this.
If you could only believe the Maths of this, it would just 'fall out'. Arm wave can never be as convincing because the description is really not adequate.
If you take two sinusoidal signals, V₁ = A sin(ω₁t) and V₂=A sin(ω₂t) and subject them to a non linearity, say a simple square law where
V_out =V_in²
= (V₁ + V₂)²

Write that in full V_out = (A sin(ω₁t) + A sin(ω₂t))² then expand it and follow @CWatters idea of using trig identities.
You get:
V_out = A²((sin(ω₁t)² + (sin(ω₂t)²)+2sin(ω₁t sin(ω₂t))
The left hand two terms will contain 2ω₁ and 2ω₂ and the right hand term will contain terms with ω₁+ω₂ and ω₁-ω₂ which corresponds to second harmonics of the input signals and sum and difference frequencies.

You say "subject to nonlinearity" and then do nonlinear maths. That's just maths which don't help me imagine at all.

I know frequency doubling is a nonlinear process but how do I imagine two waves becoming one with double the frequency? I still don't understand why the two waves don't inerfere (superimpose) as usual.

I'm trying to visualize in my head two EM waves propagating in 3d space and just can't see the frequency doubling. In my imagination waves always interfere (changes in amplitude but not in frequency).

 

[Charles Link]

You say "subject to nonlinearity" and then do nonlinear maths. That's just maths which don't help me imagine at all.

I know frequency doubling is a nonlinear process but how do I imagine two waves becoming one with double the frequency? I still don't understand why the two waves don't inerfere (superimpose) as usual.

I'm trying to visualize in my head two EM waves propagating in 3d space and just can't see the frequency doubling. In my imagination waves always interfere (changes in amplitude but not in frequency).

In the vacuum, everything is linear. This phenomenon does not occur in the vacuum. If you write the terms out classically (as J.D. Jackson does in his Classical Electrodynamics textbook), a $\dot{P}$ source term, (radiating dipole), in a non-linear material has a $\cos(2 \omega_o t)$ term if you do a Fourier transform. For a more in-depth treatment, Boyd's Non-Linear Optics book is a very good one. $\\$ The above assumes $P=\chi_o E+\chi_1 E^2$, etc.

 

[RPinPA]

Are you aware of Fourier theory, where you break a periodic signal down into a fundamental frequency and its harmonics?

If you have nonlinearities, a sine wave can be distorted. If it's not a pure sine wave of frequency f, it has harmonics 2f, 3f, 4f etc. Simple as that.

 

[sophiecentaur]

That's just maths which don't help me imagine at all.

When you are working out the cost of three ice creams, you would not, presumably, just use your imagination. You would use Maths (Arithmetic). There is no substitute for maths in everyday life. Some Maths is just harder than other Maths. Without using Maths, there are some things that you will just never understand. It's worth while getting used to 'more Maths'.

 

[Klystron]

How can two waves interacting with each other produce a wave with double the frequency? I always thought two waves can only interfere constructively or destructively, hence, increasing or decreasing their amplitude but keeping frequency the same.

The above posts applying math and trig supply solutions to the question.

For a physical example of harmonics sans math look inside a piano either vertical or horizontal (grand piano). Each key connects to a hammer that strikes a collection of taut strings that vibrate together around a central frequency to form a 'note'. Strike hard enough increasing 'loudness' or amplitude and one might notice strings associated with other notes vibrating although their hammer was not struck. You can see, hear, even feel the harmonic frequencies build, reinforce and fade.

A classical piano turner uses a reference frequency in the form of calibrated tuning forks to adjust string tension to produce the 'center note' and desired harmonics. Still doubtful? Listen to a vibrating tuning fork held in the air then touch the base to dry wood. The tone, faint in air, becomes robust. What has happened?

 

[DariusP]

Are you aware of Fourier theory, where you break a periodic signal down into a fundamental frequency and its harmonics?

If you have nonlinearities, a sine wave can be distorted. If it's not a pure sine wave of frequency f, it has harmonics 2f, 3f, 4f etc. Simple as that.

But a second harmonic is a pure sine wave just with 2f and it was made (by conservation of energy) from two pure sine waves with frequency of f.

It is a wave. Not a complex wave (a mix of different frequencies) like in Fourier theory.

Complex waves don't have a periodic amplitude. Only if you add 1f and 2f can you get a distortion but I don't see how it explains making of one 2f out of two 1f.

The green wave is a superposition of blue and red. If I combined two blues I wouldn't get red. I'd get interference... frequency wouldn't change...

 

[Charles Link]

But a second harmonic is a pure sine wave just with 2f and it was made (by conservation of energy) from two pure sine waves with frequency of f.

It is a wave. Not a complex wave (a mix of different frequencies) like in Fourier theory.

Complex waves don't have a periodic amplitude. Only if you add 1f and 2f can you get a distortion but I don't see how it explains making of one 2f out of two 1f.

View attachment 235811

The green wave is a superposition of blue and red. If I combined two blues I wouldn't get red. I'd get interference... frequency wouldn't change...

That's why you need a non-linear medium to generate the 2f. And meanwhile, you do need to be somewhat careful using Fourier theory in non-linear systems, but the non-linearities can provide enhancements to the system, (e.g. some of the linear response integral equations no longer apply, and thereby the basic convolution theorems also no longer work). You can get harmonics generated when the source started out simply as a fundamental frequency. $\\$ Additional item: You have studied simple harmonic motion where the force from a spring is $F_s=-kx$ and also when you include a driving force $F_D=A \cos(\omega t)$. If the force from the spring is $F_s=-kx+bx^2$ and Hooke's law no longer holds, the steady state will still be at the fundamental driving frequency, but you will introduce harmonics to the response function $x=x(t)$. The differential equation is $m\ddot{x}+\gamma \dot{x}=F_s+F_D$, where $\gamma$ is the coefficient for any velocity dependent damping forces. $\\$For the non-linear optical medium, the incident electromagnetic wave supplies the driving force, and the non-linear material provides the response that generates harmonics of the driving frequency.

 

[sophiecentaur]

But a second harmonic is a pure sine wave just with 2f and it was made (by conservation of energy) from two pure sine waves with frequency of f.

This is no more than an 'assertion'. If you add two sinusoidal signals (using, say, the currents in a resistor) then you will just get another sine wave with an amplitude that will depend on the amplitudes of the two waves and their phases. Another relevant factor will be the Impedances of the two sources - which will affect each other. That's usually ignored when someone tries to show that something "can't happen". In the case of two waves ( one, two or three dimensional) the result will also be sinusoidal vibrations at the original frequency with a spatial pattern which will be partly a standing wave and partly a progressive wave, depending on the details.
To generate a second harmonic (or any other product, other than the fundamental), you need a non linearity.

 

[RPinPA]

Here's a simple example of a nonlinearity that generates this sort of thing. Let's suppose you have an amplifier which is not quite linear. You wanted it to produce output $g(t) = 10f(t)$ from any input $f(t).$ But instead the response is slightly curved, and what you have is $g(t) = 10f(t) + 0.1f(t)^2$. What happens when $f(t)$ is a pure sine wave? What does that square term produce?

Let $f(t) = \sin(\omega t)$.
Then $f(t)^2 = \sin^2(\omega t) = 0.5 - 0.5 \cos(2\omega t)$ by the identity $1 - \cos(2x) = (\cos^2 x + \sin^2 x) - (\cos^2 x - \sin^2 x) = 2 \sin^2 x$

So the output of the amplifier is $g(t) = 10 \sin(\omega t) - 0.05 \cos(2 \omega t) + 0.05$. It has a component of twice the original frequency, produced by the nonlinearity. It also has a small DC term but we don't care about that.

Nonlinearities produce harmonics. If you have a channel, a medium, where the response is not proportional to the input, you will get harmonics that weren't originally there.

If you add two 1 V sine waves and your medium is linear for 0-1V, but starts to get nonlinear beyond +-1.5 V, then when your two sine waves add up the output will not be a pure sine wave. It will have harmonics.

 

[boneh3ad]

Superposition is a linear phenomenon. It involves the linear combination of waves that can then constructively or destructively interfere with one another to produce a new waveform. Harmonics are nonlinear phenomenon.

If the governing equations for a given wave include only linear terms, then only superposition can occur. If they contain nonlinear terms, then nonlinear behaviors such as quadratic phase coupling (the generating process for harmonics) can occur. Most situations contain linear and nonlinear terms, so both can occur. Usually the nonlinear terms are small, so they only matter when the amplitudes are large.

 

[sophiecentaur]

For a physical example of harmonics sans math look inside a piano either vertical or horizontal (grand piano).

The modes of vibrating steel strings are fairly well behaved but piano strings are not isolated and the situation is not steady state. Strings are tuned to a common scale (even tempered) and the different high order modes of many pairs of strings with different notes will coincide closely. This allows 'unintended' forced oscillations (one of the reasons for the felt dampers on each string). The spectrum of the sound of the 'struck' note on a piano is always very complicated and changes dynamically. Added to that, the system is not highly linear, I think.

It would perhaps be better to look for this effect in pipe organs where the notes are continuous and the spectrum is relative simple, compared with the piano. I don't know of any such experiments but they could be interesting.

Here's a simple example of a nonlinearity that generates this sort of thing. Let's suppose you have an amplifier which is not quite linear. You wanted it to produce output g(t)=10f(t)g(t)=10f(t)g(t) = 10f(t) from any input f(t).f(t).f(t). But instead the response is slightly curved, and what you have is g(t)=10f(t)+0.1f(t)2g(t)=10f(t)+0.1f(t)2g(t) = 10f(t) + 0.1f(t)^2. What happens when f(t)f(t)f(t) is a pure sine wave? What does that square term produce?

Unfortunately, the OP seems to want a Maths-free explanation. A bit like asking for a hand waving reason for how many beans make five beans.

 

[Klystron]

The modes of vibrating steel strings are fairly well behaved but piano strings are not isolated and the situation is not steady state. Strings are tuned to a common scale (even tempered) and the different high order modes of many pairs of strings with different notes will coincide closely. This allows 'unintended' forced oscillations (one of the reasons for the felt dampers on each string). The spectrum of the sound of the 'struck' note on a piano is always very complicated and changes dynamically. Added to that, the system is not highly linear, I think.

It would perhaps be better to look for this effect in pipe organs where the notes are continuous and the spectrum is relative simple, compared with the piano. I don't know of any such experiments but they could be interesting.

Unfortunately, the OP seems to want a Maths-free explanation. A bit like asking for a hand waving reason for how many beans make five beans.

Impressive analysis based on imperfect inputs, SC:

1. The piano example was only for convenience, figuring OP more likely to have access to a piano.
2. My initial mental model was the giant pipe organ at a chapel behind old mission Santa Barbara.
   
3. Yes, damping follows 'naturally' from the organ (and building!) construction without the contrived dampers in harps and pianos.
4. Try ignoring harmonics when activating one organ note causes air flow through banks of tuned pipes that rock the house! That is, induce vibrations in surrounding materials.
   
5. Electronic sources are easier to filter and amplify but even my finest maser 'quavered' and tended to phase shift from center frequency albeit a 'few octaves' above typical pipe organ resonance frequencies. Thanks.

 

[sophiecentaur]

The piano example was only for convenience, figuring OP more likely to have access to a piano.

That's an interesting post; someone with organ experience was there!
The trouble with quoting real situations is that there are already harmonic components - or at least overtones and all you need to do is 'tune' for them and up they come. I'm not sure that the OP would appreciate the vast difference between that and starting off with just a sinusoid or two.

Electronic sources are easier to filter and amplify

Absolutely. At least you have a chance of knowing what your are starting with. I haven't had any experience of Masers. They have a raw deal these days as they were in there first, I believe, but people probably choose to call them "Microwave Lasers". I remember an early book with a title "Optical Masers" by O.S. Heavens. I had a copy of it once.

 

[sophiecentaur]

based on imperfect inputs,

Could you expand on that?

 

[Klystron]

Could you expand on that?

On imperfect data:

Reading through posts including beginner-level homework and initial posts by new members, I'm impressed by the problems facing mentors.

• Posters select the subject, sub-forum, knowledge-level, thread title, and initial content. All require correction.
• language skills vary. Word choice, keyword selection, spelling, grammar, common word usage, other variables; influence understanding. What is the poster trying to write? What is the relationship between apparent written language skills and knowledge of the subject?
• Members knowledgeable in one or more fields face similar classification problems responding to the original posts. Mentors must manage myriad messages. Playful posters possibly provide pernicious proofs. [Pardon me.]

Even after successful primary classification additional posts alter the stream. Helpful members provide physical examples, analogies, pet explanations; that require rapid analysis. Is the suggested example understandable relevant, factual, ontological, logical, meaningful and useful to the thread?

Subject: wave interactions.
Measurements: frequency, wave forms.
Analogs: music (sound waves), electronics (sine wave functions)
Examples: music (harp, piano, pipe organ), electronics: (oscillators, RF cavities, maser?)
Notes:
Trying to model the apparent scope of the thread, I suggested observing a piano playing a single note. You point out that piano notes are mechanically dampened by default (unless certain pedals are pressed) and that a note from a pipe organ provides a more accurate model.

 

[sophiecentaur]

On imperfect data:

Reading through posts including beginner-level homework and initial posts by new members, I'm impressed by the problems facing mentors.

• Posters select the subject, sub-forum, knowledge-level, thread title, and initial content. All require correction.
• language skills vary. Word choice, keyword selection, spelling, grammar, common word usage, other variables; influence understanding. What is the poster trying to write? What is the relationship between apparent written language skills and knowledge of the subject?
• Members knowledgeable in one or more fields face similar classification problems responding to the original posts. Mentors must manage myriad messages. Playful posters possibly provide pernicious proofs. [Pardon me.]

Even after successful primary classification additional posts alter the stream. Helpful members provide physical examples, analogies, pet explanations; that require rapid analysis. Is the suggested example understandable relevant, factual, ontological, logical, meaningful and useful to the thread?

Subject: wave interactions.
Measurements: frequency, wave forms.
Analogs: music (sound waves), electronics (sine wave functions)
Examples: music (harp, piano, pipe organ), electronics: (oscillators, RF cavities, maser?)
Notes:
Trying to model the apparent scope of the thread, I suggested observing a piano playing a single note. You point out that piano notes are mechanically dampened by default (unless certain pedals are pressed) and that a note from a pipe organ provides a more accurate model.

I do understand that and it's right that educative threads should be tailored carefully. One thing a beginner-lever questioner should be told it that there is often not an arm waving answer available. When an analogy is used, it really has to be bomb proof and the overtones of a piano (or any musical instrument) are not the result of "frequency doubling". They exist already, due to the modes of the string and where it has been struck, whether or not the piano is treated as non-linear.

If you want a 'bomb proof' example of frequency doubling then the best one I can think of is (still a non linear process, of course) pushing a child on a swing on every other oscillation (half rate) -or even on every third or fourth oscillation. It works because the applied force is not sinusoidal but in short pulses, containing many existing harmonics. Introducing Fourier into a basic level thread is actually risky and can end up in long term misconceptions about the nature of oscillations unless it's done suitably.

The piano - organ parts were interesting but somewhat red herrings.

Loved your alliterations, btw and PF promotes pernicious proofs often. (Is this going to become a habit?)

 

[Klystron]

Thanks for the guidance. While amusing in proper context, alliteration can be tedious and superficial.
The swing-set pumping analogy also illustrates aspects of masers rather well.

 

[rude man]

How can two waves interacting with each other produce a wave with double the frequency? I always thought two waves can only interfere constructively or destructively, hence, increasing or decreasing their amplitude but keeping frequency the same.

If by 'acting' you mean addition, as in an interferometer setup, then you are correct. There is no new frequency generated, not double or anything else.
Where did you get that idea?

 

[.Scott]

How can two waves interacting with each other produce a wave with double the frequency? I always thought two waves can only interfere constructively or destructively, hence, increasing or decreasing their amplitude but keeping frequency the same.

How about this:
I will use a piano string. I will hit the string once - bringing it to a very high amplitude. Then I will strike it a second time - in phase. So the sum of the energy from the two strikes are additive. The string will now be vibrating at an amplitude so high that it will strike against other parts of the piano - creating a non-linear effect that will sound like a buzzing. The frequency of that buzzing will include many harmonics to the original frequency - including the second harmonic, double the frequency.

So long as it is simply a vibrating string, the effects are linear. Once the string reaches an amplitude where it collides with other parts of the piano, the effects become non-linear.

 

[sophiecentaur]

How about this:
I will use a piano string. I will hit the string once - bringing it to a very high amplitude. Then I will strike it a second time - in phase. So the sum of the energy from the two strikes are additive. The string will now be vibrating at an amplitude so high that it will strike against other parts of the piano - creating a non-linear effect that will sound like a buzzing. The frequency of that buzzing will include many harmonics to the original frequency - including the second harmonic, double the frequency.

So long as it is simply a vibrating string, the effects are linear. Once the string reaches an amplitude where it collides with other parts of the piano, the effects become non-linear.

We have now moved on from the OP's question. A vibrating string introduces resonance and overtones into the discussion and they have nothing to do with the basic topic of frequency addition or generating harmonics.
If you hit a piano string it will start off with many overtones. This is a transient effect. When you strike it again you will change the mode of oscillation but it's really too complicated to bother with as a thought experiment. (at least not for starters).
Better: If you excite it with a sinusoidal wave then it will resonate at its natural fundamental frequency if you get it right. A high enough level of excitation can make its amplitude large enough to, as you say, strike other parts and that can transfer some of the fundamental energy to other, higher modes.
An even better system to consider to start with is a simpler one like a mass on a spring which has just one mode. Now hang another mass and spring with twice the natural frequency next to it on a slightly flexible mount. No energy will (should) be transferred to the second spring until you force large amplitude vibrations on the first spring and it hits the stop. Then the second (or other) harmonics will be generated and will couple to the faster mass-spring. Not so familiar a model but you can apply it in a way that actually deals with the original question and avoids other distractions.
Physics is always better discussed with the simplest possible models if you want to get good understanding and not get into endless "what if?" type distractions.