What are harmonics and what kind of wave creates them

[Peter Walker]

Harmonics are multiples of any given frequency. For example if you have a frequency of 1000 Hertz then 2000 Hertz is the second harmonic of it, 3000 Hertz is the third harmonic etc. Normally harmonics are integer multiples - although they don't absolutely have to be. Fractional harmonics are very rare however.

Sub harmonics are divisions of any given frequency. So 500 Hertz may be considered to be a subharmonic of 1000 Hertz. But the definition of subharmonics is more complicated than straight harmonics and we don't normally need to consider them.

Harmonics are sometimes also called "overtones" and subharmonics are sometimes called "undertones".

A sine wave is considered to be the purest form of wave - it contains only one frequency and no harmonics. There is a mathematical theorem called Fourier's theorem that says that any wave of any shape or form or harmonic content may be made up just by using lots of different sine waves. Engineers use Fourier series (and other mathematical tools called Fourier transforms) to work out how to break any non-sine wave into its harmonics for analysis. The harmonic content of any wave is called its spectrum.

There is a simple rule that allows one to work out if any wave contains odd or even harmonics - quite simply, you draw a horizontal line through the centre of the wave, and if the wave is symmetrical above and below the line, then the wave contains only odd harmonics. If the wave is not symmetrical in this way then it must contain even harmonics (but may also contain odd harmonics at the same time).

Any wave, no matter what the shape or form or frequency that is not a pure sine wave contains harmonics.

A square wave is a wave that is basically rectangular in shape. The true definition of a square wave is that the time the wave is low is equal to the time the wave is high (see Duty cycle for more on this). A square wave consists of a fundamental (its basic frequency) and an infinite number of ODD harmonics. For example, a 1000 Hertz square wave will contain a 1000 Hertz sine wave, plus a 3000 Hertz sine wave, plus a 5000 Hertz sinewave and so on right up to infinity. Each of the harmonics will be reduced in amplitude (size or intensity) by the ratio of their harmonic number - what this means is that the 3rd harmonic (3000 Hertz in the example above) will be only 1/3 rd of the amplitude or strength of the fundamental, 1000 Hz. The 5th Harmonic (5000 Hertz) will be 1/5th of the amplitude or strength of the fundamental.

A triangle wave is like a straight ramp (diagonal) that goes up and then comes down again at the same rate (also diagonal) - imagine a triangle! It's symmetrical. A triangle contains only ODD harmonics just like a square wave except that the harmonics drop off in intensity much more rapidly than a square wave.

A sawtooth wave looks like one of the teeth of a saw! It's a straight ramp (diagonal) that goes up and then suddenly drops straight down to zero. It is not symmetrical. A sawtooth contains both ODD and EVEN harmonics. The harmonics drop off by the ratio of their harmonic number (like the square).

A damped wave (for example a damped sine) is a wave that starts off at one level and decreases in intensity over time.

There are many other wave forms and shapes possible but these are the main ones encountered in Rife type work.

[Louis Palmer]

Hi Peter,

I am a little confused! When you define a square waves you said that it would contain a 1000, 3000 and 5000 hz etc. sine wave. I generated the 1000 hz tome with my music editor and analyzed it with the freq. analyzer. I found the 1000, 3000, 5000 hz in the right ratio but they did not look like sign wave. In fact it look like a very good square wave. What am I missing?

Spiritlrp

[Chris Carthel]

Louis:

I recently did some plotting (attached) in Microsoft Excel involving Fourier analysis. A square wave is the SUM of the fundamental (sine) frequency and its odd order (sine) harmonics. As you SUM the fundamental with greater numbers of its harmonics, the resulting waveform becomes progressively more square in shape.

You might find it of interest to enter the search term "square wave" at the following link:

http://mathworld.wolfram.com/

Chris

[Louis Palmer]

Hi Chris,

That makes things a lot clearer. Some where I read that a square wave only used the positive parts of the wave and never went below zero. That created the confusion.

The link you sent is great.

The freq. analyzer in Cool Edit show the main freq. as well as all the harmonics and in most cases it does show the odd harmonics decreasing in amplitude as predicted but not at all freq. At some freq I see the even harmonics creeping in and increasing in amplitude. I am trying to make a file of the charts so I can post some of it. The problem is it is inside the application.

Thanks again, you hit the nail right on the head.

Spiritlrp

[Louis Palmer]

Hi Chris,

Been thinking about your response, I to have Microsoft Excel, how did you do the Fourier analysis? Do you think it might work on an mp3 file? I have been able to generate a mp3 file of the zapper on my system but what I think is noise is so complex I could not tell what the freq. are. Either its noise or there are so many freq being generated that its a mess. I don’t know what the manufacturer programmed in, I know it is not the standard HC as the system only goes to 10200hz.

Spiritlrp

[Chris Carthel]

The Fourier analysis I did was a plot of x(angle) versus sin(x) and its sum with successive iterations of (1/n)sin(nx), where n represents the odd order harmonics 1,3,5,... . For example, the first harmonic would be represented by (1/3)sin(3x). There are MANY fourier series. I was simply plotting the most basic mathematical concept of a Fourier series in an effort to enhance my own understanding. Learning the math behind these types of waveforms is very enlightening.

What you are referring to is basically the field of harmonic analysis.

In order to analyze an MP3 file, I presume you would need a method of converting the signals (from a transmission or recording) into data points (e.g., a spreadsheet) for analysis. I currently don’t know how to do this. You would then need to perform harmonic analysis of the data. Although I recently completed the calculus series through differential equations, this is a VERY complex mathematical subject with which I do not currently have a full grasp. Unfortunately, Fourier series were not covered in the Calculus III course I attended, so I am trying to learn more about them on my own.

I am sure there are software programs in existence that can perform harmonic analysis. But this can be complicated by the fact that signals in the real world may also be carrying distortions, interference, over-amplifications, etc. that will make the overall signal very difficult to analyze. So the numerical analysis may end up involving approximations at best. Do you know what fundamental frequency is claimed for the signal you are trying to analyze, or is it of a totally unknown nature? If you knew what fundamental frequency was claimed, then you could approach this from the opposite direction. You could (with a synthesizer for example) attempt to produce a square wave version of that frequency to see how close it matches.

Mathworld.wolfram.com states "the computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. . . . In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytic solutions" < http://mathworld.wolfram.com/FourierSeries.html>.

[Louis Palmer]

Hi Chris,

Thanks for your reply and info. The work you did is very good and makes understanding a square wave very easy. I know there is a way to take a raw set of data and perform a Fourier analysis but I am sure it involves equipment I don’t have and is most likely to expensive to buy for one person. I did get a digital multimeter 22-881 from Radio Shack the will permit me to measure freq., electric fields and a host of other things. It was well worth the 50 bucks.

I am in contact with the people that I purchased the unit from in N.Z., unfortunately they are just selling the system and have gone back to the manufacture with my request. This will take time.

The system I have has many functions, one of which is a brain tuner BT9. I was able to analyze it through Cool Edit and found many freq. ranging from 100 hz to over 20 khz. In intervals of about 100 hz. This is interesting as the unit is only suppose to generate 10.2 khz. As I later found out from other sources the freq. is not the important factor with BT systems the pulse is. These systems through out over 500 freq. in an electrical pulse at 111 cycles per second or 111 hz. Through the use of CE I was able to calculate the pulse rate. Things sure would be a lot easier with a manual but that is one of the short comings of this deal. The unit works well, has a lot of functions and is priced right but there is very little info. I am new to Rife so the black board is blank and needs to be full.

Many moons ago I use to teach DE but I am so rusty it some times hurts. Thanks for all your help.

Spiritlrp