Talk:Subharmonic

Self-contradictory article
One paragraph states (correctly, I think) that the subharmonic series is comprised with ratios of 1/n relative to the frequency of the oscillator. The next paragraph states that it is a misconception and describes combination tones. Perhaps this term has multiple meanings. Sergivs-en (talk) 03:45, 18 January 2012 (UTC)

Moved comments by 63.231.234.97 from the article text to here
Comments by 63.231.234.97 ( in brown ) on 25 January 2012‎ moved from article to here: ( by Crowsnest (talk) 09:29, 26 January 2012 (UTC) )

In music and dynamics, subharmonic frequencies are frequencies below the main frequency of a signal.

For a pure tone – of a single frequency – subharmonics occur in a ratio of 1/n to the fundamental frequency of an oscillator, with n a positive integer number. For example, if the fundamental frequency of an oscillator is 440 Hz, sub-harmonics include 220 Hz (1/2) and 110 Hz (1/4). Thus, they are a mirror image of the harmonic series.

A different usage is for the case of several fundamentals, (most often) with frequencies close to each other. Then the subharmonics are located at the difference frequencies of the original fundamentals.

(The author and later editors seem to have confused wavelength and frequency, the only way to produce f/n frequencies from a pure source is with the usage of a frequency divider and/or active feedback circuits. These frequencies should not be confused with naturally occurring subharmonics which never occur at f/n frequencies unless modulated with, for example, a frequency of 3/2*f > 3/2*f - f = 1/2*f which is clearly a difference frequency . The point of my original edit and reference was to inform the community of a common fallacy perpetuating among academics, that is the above assertion that subharmonics always occur at quantized n integer multiples of f/n. In most physics texts, that explain harmonics, the nodes are located at a boundary that is normally held constant to show that the harmonic series occurs only at n integer multiples; in the real world this is almost never the case, especially at the quantum level.)

Subharmonics are well known and form part of the combination tones, as they are called by physicist Hermann von Helmholtz and were formally described by musician Giuseppe Tartini. In fact the harmonic series is also the set of sums and differences, the only occurrence of integer multiple harmonics is with the case where a single sinusoid interacts with itself or its own harmonic.

Subharmonics can be produced by signal amplification through loudspeakers. They are naturally produced by bells, giving them their distinct sound.

String quartets by composers George Crumb and Daniel James Wolf as well as works by violinist and composer Mari Kimura require string instrument players to bow with sufficient pressure that the string vibrates in modes with frequencies corresponding to subharmonics. It is also possible to produce several subharmonics on the saxophone in a similar way to producing regular harmonics.

(Actually this is not completely correct. The string itself does not vibrate in a lower frequency mode. It is only with the modulation of two separate frequencies and the subsequent demodulation occurring in the human ear canal, in the resonating horn of the instrument, or simply in the space in which the frequencies are demodulated that subsequent subharmonics will be realized. For all practical purposes the vibrating string does indeed come close to being held to the condition of bounded nodal points and thus for the most part the string by itself mainly produces integer multiple harmonics. In the case of a horn instrument this also applies to the reed or lips.)


 * The assertion of subharmonics occurring only at ratios of 1/n is absolutely false. ANY ratios can be produced the driving factor IS AND ONLY IS the difference between the frequencies of the two or more combined frequencies. If the editor would do some research into the world of frequency synthesis he might become enlightened.  — Preceding unsigned comment added by 63.231.234.97 (talk) 14:38, 26 January 2012 (UTC)
 * Hello, IP editor. It would seem that you know what you are talking about, which, I freely admit, is a lot more than I do. What is needed here to start sorting this article out is some reliable references; Wikipedia articles are based on published sources, not on the knowledge of contributors, however expert. Can you point to books or academic articles that support or explain what you are saying? Speaking purely for myself, as my knowledge of this topic is a long way south of zero, it would be particularly helpful if some of those sources could be viewed online. Justlettersandnumbers (talk) 18:29, 26 January 2012 (UTC)

I would love to and I can do some research to provide some addition ones. Thank you, I thought that I was about to engage in a pile on, apparently this is not the case and my view of WIKI as a reliable source may have just been saved. I am not nor do I wish to become a WIKI guru which is why I have not created an account.

The problem here is that there is an enormous amount of academic pressure to support anything that props up quantum theories inside the world of physics. This includes any assertions of "n" integer anything. (there is another paper of mine that addresses this issue, a link to this paper is also on my home page www.photodady.com) There are in fact a lot of other sites, including links to other wiki articles that I used in this article, that also promotes the fallacy of integer multiple harmonics. The lies are that of the omission of any other harmonic frequencies. I personally think the bias among academia stems directly from the academic bias against Nikola Tesla, the person who actually first understood and used the assertions that I have made. Academia simply has not incorporated Tesla's research in their introductory texts, how unfortunate since in the practical world of radio they are utilized extensively and often contradict academic texts. This is most likely where I will begin to research for more citations for you pleasure. — Preceding unsigned comment added by 63.231.234.97 (talk) 19:01, 26 January 2012 (UTC)

Well here is one reference to a very similar topic that is addressed in nearly every introductory level physics text, it is called "beat frequencies." The sad part about this is the trivial way in which this topic is treated. Beat Frequency = |f1 - f2|, this is one of the only references that I can find that related to difference frequencies in my physics texts. The real question here is: Why this is topic is treated so trivially? Especially when there are such important wide ranging applications and such disinformation propagating because of the lack of applying it. (I have my theories) The text is: Physics for Scientists and Engineers, 6th Edition by Serway and Jewett pages 564-565. (By the way photography is just one of my disciplines, I am also a licensed FCC radio technician, FAA licensed aircraft technician, FAA licensed private pilot, I also possess an electrical engineering bachelor degree.)

I will continue researching for even more references for your delight. — Preceding unsigned comment added by 63.231.234.97 (talk) 21:59, 26 January 2012 (UTC)
 * Thanks for your reply. A couple of very brief comments: if the theory of subharmonics was treated by Tesla, then some links or references to his treatment would be useful. The physics book you cite (well, actually I found a different edition on Google) discusses beat frequencies, a phenomenon well known to all (?) practising musicians, as they are a fundamental tool in tuning and intonation. And thirdly, I'm sorry, but the personal theories of you or any other editor, however interesting, have no place here. Wikipedia is based on published reliable sources, not on individual theories. I look forward to further references. Justlettersandnumbers (talk) 23:20, 26 January 2012 (UTC)

After looking at my textbooks I did a couple of searches and came up with a plethora of applications ranging from NASA for laser generation to the medical field in ultrasound imaging to basic radio principles all by using the frequencies that fall under the category of subharmonic or difference frequencies. Here are just a few: (Also NONE of the content of my paper referenced for this article are of my own personal theories, that was the entire point.)

http://www.google.com/search?client=safari&rls=en&q=difference+frequencies&ie=UTF-8&oe=UTF-8#pq=difference+frequencies&hl=en&sugexp=pfwl&cp=7&gs_id=r&xhr=t&q=modulation&pf=p&sclient=psy-ab&client=safari&rls=en&source=hp&pbx=1&oq=modulat&aq=0&aqi=g4&aql=&gs_sm=&gs_upl=&bav=on.2,or.r_gc.r_pw.,cf.osb&fp=5c70f6dd121aab58&biw=1801&bih=1055

http://pra.aps.org/abstract/PRA/v75/i4/e041801

http://adsabs.harvard.edu/abs/2009PhRvA..80b1804O

http://www.rp-photonics.com/sum_and_difference_frequency_generation.html

http://michaelgellis.tripod.com/mixersin.html — Preceding unsigned comment added by 63.231.234.97 (talk) 00:31, 27 January 2012 (UTC)

Here is a reference from the Acoustical Society of America describing how the subharmonic series of a violin is related to the characteristics of the wood as well as the front and back plates (the violins resonating horn, A.K.A. the Helmholtz Resonator):

http://asadl.org/jasa/resource/1/jasman/v32/i11/p1443_s1?isAuthorized=no

If you can't accept the above sources as supporting my input towards correcting the fallacies that I found in this article then I am more than sure that I can properly interpret your true intentions here. — Preceding unsigned comment added by 63.231.234.97 (talk) 03:02, 27 January 2012 (UTC)

Unreliable reference?
I have removed the references to an "original paper" by this photographer. Unpublished original research cannot by any stretch of the imagination be regarded as a WP:Reliable source. Justlettersandnumbers (talk) 18:19, 26 January 2012 (UTC)
 * And removed it again, after requesting confirmation at RSN that my initial judgement was correct. Please do not add it back to the article without providing convincing evidence that the author, Daniel Joseph Dady, is "an established expert on the topic of the article whose work in the relevant field has previously been published by reliable third-party publications". Specifically, that means citations of articles by this author, in this field, in reliable peer-reviewed academic publications. In the absence of those, this source cannot be regarded as reliable, and has no place in Wikipedia. Justlettersandnumbers (talk) 22:57, 26 January 2012 (UTC)


 * The reason I have not chosen not to publish my paper in a "peer reviewed" publication is because many of the publishers, as well as the peer review processes, are well known to be corrupt. For example just research the "peer reviewed" articles surrounding human caused global warming that have widely promoted this fallacy. Also very recently a large group of scientists have actually openly boycotted one publisher, see article in The Chronicle:
 * http://chronicle.com/article/As-Journal-Boycott-Grows/130600/


 * Also note that it was the "peers" who had placed Galileo under house arrest for daring to suggest that the Earth was not the center of the universe. Thank God that Copernicus chose not to publish his work in a "peer reviewed" article.


 * Please if there is any criticism or fallacies that you might have found in my papers I'm anxiously awaiting your reply.
 * 63.231.234.97 (talk) 15:02, 1 February 2012 (UTC)


 * Interesting find, an "expert" whom I have worked with in the past, and expressed my doubts, has two papers submitted for "peer review" and subsequent publication concerning "heterodyning" (a radio term) in order to predict and control stability in machining operations through the use of "beat frequencies."
 * I have not read these papers but a listing of their existence can be found:
 * http://mtrc.mae.ufl.edu/schmitz.php?publications
 * 63.231.234.97 (talk) 17:57, 8 February 2012 (UTC)

Subharmonic = beat frequency?
Someone please clarify this. In this context, does the term subharmonic mean difference / beat frequency? Does the term ever refer to f/N? It is clear that such f/N do not occur naturally in the spectrum, but the name does suggest something like that. Sergivs-en (talk) 14:41, 30 January 2012 (UTC)


 * Mathematically speaking sum and difference frequencies are the direct result of applying sum to product trigonometric identities when two sinusoids are superposed or added, see List_of_trigonometric_identities. To answer you question; practically speaking sum and difference frequencies are one in the same as beat frequencies. The term beat frequency is usually used when the subharmonic frequency is below the audible spectrum and is heard by the swelling or changes in volume of the signal. Beat frequencies are used for tuning string instruments as well as in radio with the use of Beat_frequency_oscillators or BFO's which were once widely used for determining the proper tuning of a radio signal in a radio receiver.63.231.234.97 (talk) 18:01, 30 January 2012 (UTC)63.231.234.97 (talk) 18:03, 30 January 2012 (UTC)


 * This does not answer the question I asked. I can see that you are bursting with knowledge about the beat frequencies, but I was asking about the actual term "subharmonic" and whether it ever applies to f/N frequencies mentioned at least in the earlier version of the article, which are definitely not "one in the same" as beat frequencies Sergivs-en (talk) 09:34, 6 February 2012 (UTC)


 * So your question is one of semantics? If you are defining single pure tones that are never exposed to other tones, or their own reflections, as "natural" you might be on to something. In this very narrow and restrictive definition F/N frequencies do not exist as you have suggested, but then again neither do F*N frequencies. Scenarios where the superposition of different sinusoids naturally exist and are quite common in nature, the scenarios where sinusoids are intentionally restricted at their fundamental nodes do not fall under the term "natural" unless one has been incorrectly taught that it is so. The term SUBharmonics, as defined as harmonic frequencies below those of the fundamental frequencies, is indeed and correctly named as a SUB set of the complete set of harmonics.


 * So to answer your question a little more precisely beat frequencies are subharmonic frequencies but subharmonic frequencies are not necessarily located in the beat frequency spectrum, in so far as they have traditionally been defined and used in practice.
 * 63.231.234.97 (talk) 16:55, 8 February 2012 (UTC)


 * Just another two cents that I will add (reiterate) to the discussion here is that the consequences brought out by the fact that harmonics occur at frequencies other that F*N is that it exposes the fact that harmonics are not a in any way shape or form a discrete phenomenon, analog radio theory and practice proves it, as does the plethora of other applications that have resulted from the 1901 Heterodyne patent, that is correct I'm here defending the concepts that were patented 111 years ago. In fact digital communications will always rely on this fact. Mathematicians like having the ability to concisely contain a concept to a simple form, such as the one with the N integer harmonic series. However, I have not yet seen a concise mathematical description of the complete set of harmonics. This is a challenge to all of the mathematicians out there, in fact this is not the only area of mathematics that have been conveniently swept under the rug. I understand that it is very difficult for most to accept concepts that may differ from long standing accepted ideas and these concepts must be scrutinized very closely. Change can be difficult but ones mind must be open to corrections of past fallacies, that is the foundation of science itself. When ones mind becomes closed it can only result in pseudoscience and/or corruption by those who wish to control such information.63.231.234.97 (talk) 15:48, 9 February 2012 (UTC)


 * Quite a lot of text there but it isn't really relevant to the question, which I don't want to repeat again. Yes, it is a semantic one, I thought that was pretty clear. Sergivs-en (talk) 11:27, 11 February 2012 (UTC)


 * To repeat myself yet once again:


 * The term subharmonic is a relative term defining the spectrum that depends on the fundamental frequencies.


 * The term beat frequency is an absolute term referring the subharmonics occurring in the spectrum below that of human hearing.


 * The term difference frequency is a mathematical term that describes how harmonic and subharmonic frequencies as well as beat frequencies are formed.


 * The term sum frequency is a mathematical term that describes how harmonic frequencies, including N integer harmonics, are formed.


 * General trigonometric identity:


 * sin(A) + sin(B) = 2*sin((A+B)/2)*cos((A-B)/2)


 * For N integer harmonics let the independent term "B" equal the reflection of "A" or some N integer multiple of "A" and Shazam! No subharmonic frequencies.


 * So to reiterate more directly the answers to your questions:


 * 1a) The term subharmonic does not mean beat frequency, but beat frequencies are subharmonic frequencies.


 * 1b) The term difference frequency does not mean subharmonic, but subharmonic frequencies are difference frequencies.


 * 2)The term does refer to F/N frequencies when the difference between the superposition of two fundamental frequencies equals F/N.


 * 3)The statement that you made that "It is clear that F/N frequencies do not occur naturally in the spectrum" is categorically false.


 * Is that clear enough?63.231.234.97 (talk) 16:10, 11 February 2012 (UTC)


 * Well, not really. Perhaps my question wasn't clear and what confused the matter more is that I wrote in shorthand difference / beat frequencies. The slash did not mean division, I was simply referring to the two terms that are often used interchangeably. And I should have phrased the question "does the term ever refer specifically to f/N". Apparently, you are of the opinion that it does not. It would probably carry more weight without all the pseudoscientific gibberish. Sergivs-en (talk) 12:22, 12 February 2012 (UTC)


 * The changes the last weeks on the article reflect a very narrow view on subharmonics, as if all can be explained through simple difference frequencies. But subharmonics are an important phenomenon in dynamics: the f/N frequencies are generated through nonlinear bifurcations, like period doubling, etc. Which in cases may lead to chaos. Subharmonics occur in all kinds of physical systems and the associated mathematical models of them, not only in music. -- Crowsnest (talk) 20:45, 12 February 2012 (UTC)

"Narrow view" is putting it mildly. Obviously, the person is really excited about multiplying trigonometric functions. Thanks for the bifurcation link. It would be nice to find the definition of the term "subharmonic" in literature, though. I've seen it used somewhere before, don't remember where. The current version of the article basically says "frequencies below the main frequency of the signal". This is not a good definition, because we don't know what "the main frequency" is. A quick Google Scholar search shows "subharmonic" used with "bifurcation" a lot. So for quasi-periodic systems (?) this term is used. As far as music and acoustics, I actually have a copy of Helmholtz's "On the sensation of tones", which is mentioned in the current article (possibly by 63.231.234.97), but I've been too lazy to check whether Helmholtz actually uses the term. In any case, the article definitely needs to be cleaned up or rewritten. Sergivs-en (talk) 08:37, 13 February 2012 (UTC)


 * I just checked “On the Sensations of Tone”. The index includes “Combinational tones”, “Harmonics”, “Partials”, “Tartini”, and “Upper partials”, but “Subharmonic” is not found. The index passes from “Subdominant chord” to “Subminor fifth 5 : 7”. There’s a likely reason for that: It's an unnecessary term. By his definition, the term “partials” already includes all (not just some) of the sine components of a complex tone produced by an acoustic instrument. Only one of those can be be the lowest partial. It makes no logical sense—it’s literally nonsense—to speak of a partial lower than the lowest partial. 38.86.48.38 (talk) 17:01, 3 June 2015 (UTC)

Y Shaped Strings
Interesting take, without seeing any experimental evidence I think that this assertion may be true. The separate sections of the Y should be considered as separate independent sinusoidal sources that partially conform to the strings with bounded nodes while the center point being considered a partially bounded node with additional degrees of freedom. These strings would most likely NOT be able to produce pure single tone sinusoids but would produce superposed sinusoids directly. Similarly when an open string is lightly touched, thus reducing a degree of freedom, at the center of the string length, the fundamental will be quieted leaving only the 2nd and above harmonics. Similar effects occur when touched at other nodal points. This subject would be experimentally interesting and most likely provide substantiating evidence supporting this discussion concerning the corrections of long standing fallacies that have been, at least temporarily, corrected in this page. 63.231.234.97 (talk) 18:15, 1 February 2012 (UTC)

Another thought, by the usage of a Y shaped string the dynamics become much more complex and would come closer to the dynamics of two-dimensional vibrating surfaces, see Vibrations_of_a_circular_drum. 63.231.234.97 (talk) 18:35, 1 February 2012 (UTC)

Explanation that even a child could understand
This topic continues to be completely misunderstood by apparently very well educated persons, let me clue you into how I first arrived at recognizing this fallacy.

In short I was fishing. Take a body of water, that body of water will most likely have waves on its surface. Now lets remove any sources of energy that will create these waves with the exception of the wind. Anyone can plainly see that when a wind blows on this surface waves will form. These initial waves are low amplitude waves with a relatively short wavelength of maybe a couple off centimeters in length that can easily be observed during any wind gust, these are relatively high frequency waves. Now consider that after some time these high frequency waves will always interact to form much longer wavelengths with much higher amplitudes, these are relatively much lower frequency waves.These lower frequency waves are in fact subharmonic waves that have formed because of the differences of the superposition of the original high frequency waves.

Why is this true when during most of all of my academic experience I have been taught that waves only interact to form N integer harmonic waves? Personally I had to rely on my technical training in aircraft avionics and radio to answer this question. That is correct durrng my entire academic career in electrical engineering from a very prominent institution they completely and miserably failed to ever discuss this topic. This is a very sad commentary on our "higher" educational system.

The fact is that these lower frequency waves physically form because of both the direct mathematics already discussed here ad nauseum as well as the results of the conservation of energy theorems from thermodynamics. By the way Herman von Helmholtz is, in a large part, also responsible for this area of physics.

For a more in depth discussion on this subject please see the paper that has been previously removed from the References section but is still available in the External Links section of the main article.

http://www.photodady.com/subharmonics_rev2.pdf

63.231.234.97 (talk) 17:19, 11 February 2012 (UTC)63.231.234.97 (talk) 18:15, 11 February 2012 (UTC)63.231.234.97 (talk) 18:40, 11 February 2012 (UTC)
 * Sorry, but this is all original research, see WP:NOR on the Wikipedia policy with respect to original research. This is an encyclopedia, i.e. describing what has appeared in reliable secondary sources. Further, your observations with respect to the lengthening and growing of wind waves are completely incorrect. The mechanisms for lower frequency and higher amplitude are due to nonlinear (Benjamin-Feir) instabilities, resonant nonlinear wave interactions and interaction with the wind. See e.g. . -- Crowsnest (talk) 21:04, 12 February 2012 (UTC)

Final Thoughts
Thank you for providing collaborative research that provides credibility and independent validation to the century old research (READ: NOT MY OWN RESEARCH) that high frequency waves can and do combine harmonically to form lower frequency waves. In the area of academic research this is usually a reason to jump for joy. But I am not an academic nor was it my intention to prove this since truths always stand on their own. Proven by the incorporation and, albeit slow, acceptance of these century old radio theories into many other areas of industry. Academia continues to be the last holdout. I must admit that my intentions here were a little subversive. I was actually doing research that might help me with my long standing support for open source programming as well as open source depositories of information.

Unfortunately this is not the case, instead what I am actually accumulating is data that supports the psychological behavioral research of Alfred Adler. That people will behave irrationally and take unethical positions when acquiring and defending positions of power, virtual or otherwise. Thank you for your participation, I was surprised that it took you so long to engage in the ad hominem attacks since I personally thought that they were going to begin much sooner. In fact I thought they were going to start last August when I added my first entry to this page.

Really? I'm the one with a narrow view? Please, when it is YOU that seem to believe that academia and academic publications are the sole source of reliable truth. When yet another instance of the fallacies that have resulted in potentially accelerated human deaths that was perpetuated by numerous “peer reviewed” articles published in the highest caliber academic professional journals that was exposed last night by CBS's 60 Minutes. In my opinion yours is the narrow view.

I was hoping to prove wrong the research performed by Ron Jones and “The Third Wave” theory on fascism, I failed thanks to your help. I was also trying to disprove the assertion by Aristotle that democracies always degenerate into despotisms. Again I failed thanks to your help and unfortunatly predictable participation.

What happens in the future to the truths I provided here on this page may eventually turn around the conclusions that I am being driven towards, but I am loosing faith. Examples include: WIKILeaks as well as being driven to the Mac from the crappy and unstable MAC-WANNABE UBUNTU GUI and lack of commercial software support despite the fact that generally LINUX systems have proven to be the most stable and secure platforms available. Note: Mac OSX is a propriety branch of LINUX.

Good Luck towards all,

Especially those 3500 or so other people who have visited this page in the past 90 days but were unable to chime in due to non-disclosure agreements and security clearances.

P.S.

I'm the only one that so far who's identity is exposed for all to see. I've kept my IP address static for a reason, but not for long. It is easy to stand behind an anonymous avatar with absolutely nothing to lose but your own position in your own virtual world. That is the sad commentary on WIKI as a reliable source of truthful information.

63.231.234.97 (talk) 19:03, 13 February 2012 (UTC)


 * You're just too far ahead of your own time, and we're jealous. Sergivs-en (talk) 20:13, 13 February 2012 (UTC)


 * P.S. But seriously, please understand that this is not the place to reveal truths. I'm sorry that you're disappointed in Wikipedia, but this approach is what makes it a valuable and useful reference. Perhaps the "attacks" that you write about did not happen earlier because this is a fairly obscure subject. You obviously understand how it's supposed to work to some degree, because you write you were being "subversive". This is disrespectful to other people who are, let's say, no less intelligent than you. Either you don't think very much of their time or a little too much of your point of view. You make some points about the academic establishment, ethics, and positions in the virtual world etc. -- well, try to think clearly about your own motivations instead. And good luck with your research. Sergivs-en (talk) 02:01, 14 February 2012 (UTC)

Information that is not held to the highest standards of truth is called disinformation, thank you for clarifying WIKIPEDIA's philosophical stand on the truth. Wasn't I the fool to ever have thought otherwise. Intelligence is not he issue, one only needs to lift the blinds to be able to see the light; a willingness to be open is how this is accomplished, not through intelligence. To quote one of Nikola Teslas's close friends:

All schools, all colleges, have two great functions: to confer, and to conceal, valuable knowledge. The theological knowledge which they conceal cannot justly be regarded as less valuable than that which they reveal. That is, when a man is buying a basket of strawberries it can profit him to know that the bottom half of it is rotten.

- Mark Twain, Notebook 1908 63.231.234.97 (talk) 11:25, 19 February 2012 (UTC) I edited the main page back to a close version before I added my entry from last August, do with what I gave to you however you chose.63.231.234.97 (talk) 11:50, 19 February 2012 (UTC)

If you need a formal mathematical description of how radio modulation and demodulation actually works see:

Signals and Systems, Second Edition, Alan V. Oppenheim and Alan S. Willsky, Chapter 8 Communications

However advanced mathematics involving calculus, the complex plane, and convolution is required. The simple AM radio wave is, however, the simple superposition of three sinusoids consisting of a carrier frequency and two lower amplitude sidebands equally spaced by the modulation frequency (subharmonic frequency) above and below the carrier frequency. The trigonometric identities, already described above, dictate how these sinusoids are combined. I must warn you that if you had trouble seeing the validity of the relatively simple trigonometric identities described above then you will most likely be completely lost in this material.

One of the most disturbing things that I have noticed (even more so than this WIKI discussion) is that the single most prolific invention of the 20th century, that has had the greatest influence on the entirety of society, is radio which forms the foundation of nearly all electronic communications, including television, cable, satellite, wired and wireless internet, cell phones, bluetooth, fiber optics, GPS, and on and on and on, and continues to be completely misunderstood only because academia continues to refuse to allow the most basic of theoretical foundations from entering entry level physics texts. The concepts are actually quite simple but remain confined to advanced topics in advanced communications classes that tend to be quite literally convoluted for reasons only known by the academics who continue to want to keep it that way which leaves the average person as well as a very large segment of very highly educated people with advanced degrees completely in the dark which then allows the fallacies of this WIKI page to propagate unabated.63.231.234.97 (talk) 02:48, 20 February 2012 (UTC)