what is and why are there harmonics?

[Guest Labop]

Hi all, this is my first post. Yes I did a search, didn't find quite what I was looking for. Heck, I searched a few university/government sites that weren't that helpful. I'm basically trying to explain the why concept to my coworkers.

 

I'm a crypto tech in the US Navy. I mainly deal with RADARs. My question is to the nature of harmonics.

 

I know that a harmonic is a doubling of the original frequency. Then it doubles again, and again and so on. When you collect this on an RF pan display, you will see one spike at 1000 MHz (your signal), and another at 2000 MHz (your harmonic). They all understand the doubling factor.

 

They are to the point where they understand that anything that originates with an oscilation will have a harmonic. The original discussion was because one individual said that you will only see a harmonic if the transmitter is malfuncitoning. We've moved past that.

 

They want to know why.

 

I've tried showing the rainbow explaination (one main rainbow, a faded one to each side). I've tried drawing the sine wave as expressed as a layered circle. Nadda.

 

I've got a couple of RADAR engineering handbooks / encyclopedias here and all they do is define, not explain it. I would like some help here please!!

 

Thanks,

[Guest Labop]

*bump*

 

no takers? no one has any ideas on this one?

[Panic]

The best way to explain harmonics is to first introduce wave theory in as little space as possible.

 

you can try this at home.. If you have a clouth line or tie two string with moderate tension to from one post to another.

 

|----------------------------------------|

0----------------------------------------L

 

the lenght of this string is now length L from end to end. If you were to pop the string at end '0' you would see a wave move along the sting and then get reflected back

 

This is called wave propagation.

 

Now if you were to vibrate on end of the string with a frequency that would generate a standing wave ... meaning a wave that would apprear to not move down the string.

 

This standing wave has a specific wave legnth with relation to the disntace L

the wave will appear as 1/2 of a full sine wave with the end L fixed at zero...

see diagram below.

 

 

The frequency correstponds to the base frequency. wich has a relation to wave lenth, distance L, and angular frequency.

 

Reaching the first harmonic, from the above diagram b). is possible by either increasing the frequency maintaining the string at length L.. or increasing the string length and maintaining frequency constant.

 

a full sine wave is now called the first harmonic.

1.5 of a full sine 3rd harmonic.

2 full sine's 4th harmonic.

 

and so on and son on and so on.

 

 

Now the same theory applied to radar and sonar. although the wave no longer propagate along a string now a wave travels thru air or water. (in a different bu simmilar way)

 

another good example is with musical notes.

 

a guitar for example.

if you pluck the string scractly at the middle you will hear a more solid tone. since you are activating the base frequency "mostly"

if you pluck the string over the hole you are activating other harmonics that have a maxima or minima at that point.

 

by plucking the guitar next to the phret you will hear a meatalic twang ... that is a combination sound of multiple harmonics.

 

 

 

if you

[Panic]

I know that a harmonic is a doubling of the original frequency. Then it doubles again' date=' and again and so on. When you collect this on an RF pan display, you will see one spike at 1000 MHz (your signal), and another at 2000 MHz (your harmonic). They all understand the doubling factor.

 

They are to the point where they understand that anything that originates with an oscilation will have a harmonic. The original discussion was because one individual said that you will only see a harmonic if the transmitter is malfuncitoning. We've moved past that.

 

They want to know why.QUOTE']

 

ok now that i have explained the wave let me try to answere you question.

If you notice on the second diagram that the maximum height of the base frequency standing wave is smack dab in the middle.

 

the first harmonic has a maximum height 1/3 L and 2/3 L... but at the point where the base frequency has a Maximum it has a node... this has a lower potential to activate the base frequency then the 2nd harmonic. the second harmonic also has a maxima at 1/2 L the second harmonic has a better potential of activating the base frequncy becuase they share more comon point. both have nodes at 0 and L and both have maxima at 1/2 L.

 

 

now if your equipment is made to produce a frequency that work in the 5th of sixth harmonic of a distance for example the distance from the sonar dome of a submarine to the sear floor. well it is quit possible that you will also hear lower and higher harmonics.

 

if the emmiter of the original frequency is faulted it may fluctuate with iregular frequncy there for activating more the one harmonic.

 

a perfect example of this is a tuning fork...

 

a tuning fork has a specific frequency ... if yo uplace the fork "C" next to the guitard cord "C" the sound energy from the fork will commence to vibrate the guitard cord "C". becuase they share maximas and minamas..

 

but the "E" cord will not resonate. although they share the same distance L... the sting mass/density and tension affect the way waves trave along that sting.

[Panic]

http://www.csgnetwork.com/harmonicscalc.html

 

you can see that double frequncy does not result in next harmonic ... but it will result in "A" complemeting harmonic

[Martin]

 

They want to know why.

 

 

Panic's stuff is very good. thanks Panic' date=' the reflected traveling wave and standing wave movie is neat.

 

Labop, try them on the concept of NONLINEARITY

 

if you square a sinewave you get twice the frequency sinewave, try it.

 

 

so if you have a linear transmission function and you put in a sinewave you dont get harmonics

 

but if you have a nonlinear transmission function you get harmonics and maybe that is good (for your application) and maybe bad

 

 

example of linear f(x) = 3x

plug in sin(440 t)

f( sin(440 t)) = 3 sin(440 t), this is still a pure sinewave with no harmonics

 

example of nonlinear f(x) = 3x + (1/2) x[sup']2[/sup]

plug in sin(440 t) and you will get a superposition of something with

the same frequency plus a small amount of something with twice the frequency.

 

 

I stuck in the number 440 because it looks like a frequency but actually to get the music tone of A you need to put in 2pi times that----880pi. But this is just an example.

 

are you OK that squaring sinewave gives a wave of twice the frequency? it is a trig identity that one learns in highschool

 

cos²(t) = (1 + cos(2t))/2

[mmalluck]

Yeah Panics stuff is pretty good. Let me see if I can simplify things a bit.

 

Harmonics are the result of boundary conditions; that several waveforms can exist in a given space.

 

Lets looks at panic's second picture again. The shaded bits are wall, waveguide, post, or some other material a wave can reflect off of. The black line is your wave.

These wallls are your boundary conditions. In order to establish a standing wave (or in other words, cause this chamber to resonate) the wave can only meet the wall where it crosses zero. If the wave falls in any other manner it will fail to resonate within this given space. It turns to garbage.

 

So we start sticking waves into the cavity that meet this requirment. We find that multiples of the frequency fit nicely (.5 lamda, 1 lamda, 2 lamda......) This is true of all forms of resonance.

 

What causes this in your radar stuff? I'd venture to say that the radar is not producing a perfect sin wave. There's a theory that any wave form can be made by summing together an infinate number of perfect sine waves. This means for any wave that is not a perfect sine wave, all frequencies are represented, though many will be at a very small power.

 

In this case the resonating cavity of your radar system acts to filter out all the other frequencies (they turn to garbage), but the resonate ones, so they show up in your output.

[Guest Labop]

First, thanks for the replys, I think we're getting closer.

 

Panic, mmalluck, what you two seem to be describing is what we refer to as reflection or multi-pathing. When that occurs, you don't get a harmonic. You get a second signal at the same freq/bearing but will be slightly delayed and is usually weaker.

 

The animation appeared to be describing the combination of the original wave and reflected, and for some reason it showed a change in amplitude (height), doubling it. That does not happen with RF harmonics, amplitude would stay the same (or even drop off). The animation did not show a change in frequency (length/width).

 

Also, you don't need the starting point to measure freq, you just need two like points along the sine wave, so having the "perfect" wave isn't it either. There are no perfect waves outside of the lab.

 

Panic, using your example and applying it to what I'm talking about, if you struck the C chord you hear C and E. RF harmonics are kind of like that.

 

Let me use the rainbow example. You see a rainbow in the distance. On a "perfect" day (perfect being for the transmission on light), to each side, you see another weaker copy of the rainbow. This is all caused by diffraction of light through rain drops (another type of wave) but those rainbows to either side are essentially harmonics of the original (center) rainbow.

 

Martin's stuff is closer to what I'm getting at. Though the mathimatical explainations will only be understood by a couple of people. Martin, do you have another way of saying it in plain english? I'm trying to inform the instructors so they can better inform their students. If the instructors go "huh?" the students will to.

[YT2095]

Doppler and BFO searches may of some use also

[Martin]

... what I'm getting at. Though the mathimatical explainations will only be understood by a couple of people. Martin' date=' do you have another way of saying it in plain english? I'm trying to inform the instructors so they can better inform their students. If the instructors go "huh?" the students will to.[/quote']

 

several people at SFN can (I expect) give you a way of saying it in plain english.

I suggest you write a PM (private message) to swansont.

It is very easy to send PM at this board.

Or you can just wait and someone like swansont or severian (working physicist) may see your thread and reply.

 

I cannot say it in plain english but i can try again to say it in plain mathematics. With any NONLINEAR medium if you put in a signal without harmonics what you get out will have some harmonics or overtones.

 

harmonics can happen other ways but tell them to try to understand just this simple case: a black box where if you put in a timevaryingsignal f(t) what you get out at the other end is not pure f(t) but a modified thing G(f(t))

 

G is just some function G(x) = x + small coef x² + other small coef x³ +....

 

Let's say G is almost perfect fidelity.

Perfect would be G(x) = x, you get out exactly what you put in, and then G(f(t)) = f(t) and the signal passes thru unchanged.

 

But G has some small second order and third order terms, so it messes the signal just a tiny bit

so what comes out has a little bit of (f(t))² in it and maybe a little bit of (f(t))³ mixed in as well.

 

and when you square a sinewave you get something twice the frequency

 

try it

try squaring sin(t)

all the stuff below the t-axis gets made positive so it is above the t-axis

so now the wave has twice as many humps.

you will see. squaring a sinewave really does give you twice the freq.

 

My faith in the instructors of the USNavy is such that I believe they will not go "huh". but I can still appreciate what you are up against with the men

[Martin]

the rainbow analogy may be confusing them

the secondary rainbow is caused by extra reflection inside the drop of water (I actually do not think this is a good physical analogy to what normally causes harmonics in signal processing, but i could be wrong)

 

the second and third rainbows are somehow VISUALLY or SCHEMATICALLY similar to overtones and yet the mechanism is not quite analogous

 

take a guitar to class and touch the middle of the vibrating string to make it divide and get an octave higher pitch