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Calculus not needed (was: Reflection Coefficient Smoke Clears a Bit)
I've enjoyed reading this and related threads. Some comments have been
made about using calculus. Though I spent a significant portion of my pre-retirement life attempting to teach that subject to undergraduates, I don't believe in using calculus whenever simple geometry and/or algebra makes it unnecessary. A proof that avoids calculus can be meaningful for those who don't know calculus, or who haven't used it for a while. With that said, I have a few comments to make about some of the assertions I have read here recently, some of which have appeared without explicit proof. (1) The surge impedance of a (lossy) transmission line cannot have an angle more than 45 degrees away from the real axis. This is true. Z_0 = sqrt((R + jwL)/(G + jwC)) (here I am using "w" instead of omega). Both the numerator and denominator lie in the first quadrant, so their quotient lies in the right half plane (angles subtract when one divides), and the square root of that has an angle lying between - 45 degrees and + 45 degrees. (The branch of the square root with positive real part has to be taken; if you can find coax whose surge impedance has a negative real component, I'll pay you good money for it.) [Since this angle lies between - 45 degrees and + 45 degrees, peculiar consequences deduced from calculations involving surge impedances such as 50 - j200 can be ignored.] (2) There is a nice geometrical interpretation for the reflection coefficient, or rather for its magnitude. Since the coefficient is (Z_L - Z_0)/(Z_L + Z_0), its magnitude expresses how much further Z_L is from the surge impedance Z_0 than it is from the negative, - Z_0, of the surge impedance. If Z_L is equidistant from Z_0 and - Z_0, then the magnitude of the reflection coefficient is 1. If Z_L is closer in the complex plane to Z_0 than it is to - Z_0, then the magnitude of the reflection coefficient is less than 1. If Z_L is closer to - Z_0 than to Z_0, then the reflection coefficient's magnitude exceeds 1. Now plot the points Z_0 and - Z_0 and draw the perpendicular bisector of the segment joining them. If Z_L is on that perpendicular bisector, the magnitude of the reflection coefficient is 1; if it is on Z_0's side of the bisector, the magnitude is less than 1; if it is on - Z_0's side, the magnitude exceeds 1. Of course Z_L has to stay in the right half plane; if it didn't have to do this, you could take Z_L very close to - Z_0 and get enormous reflection coefficient magnitudes. (3) Consider an ellipse having Z_0 and - Z_0 as its foci. There are infinitely many such ellipses, including a degenerate one (just the segment between the "foci"). All these different ellipses fill up the complex plane, and no point in the plane is on more than one of them. On any one such ellipse, the sum of the distances from a point on the ellipse to the two foci Z_0 and - Z_0 is constant (definition of an ellipse), the value of that constant depending upon which ellipse it is but the constant has to be at least as large as the interfocal distance. We should ignore points on the ellipse that are in the left half plane. A portion of the ellipse will be in the same quadrant as Z_0, and a portion will be in the quadrant that contains the conjugate of Z_0. (Remember we are ignoring the points in the left half plane.) All of the points on the ellipse that are in Z_0's quadrant are closer to Z_0 than to - Z_0, so they'll give reflection coefficients with magnitude less than 1. So will those of the points on the ellipse in the other quadrant under consideration that are between the real axis and the perpendicular bisector. But those that are between the perpendicular bisector and the imaginary axis will be closer to - Z_0 than to Z_0 and thus will yield reflection coefficients with magnitudes greater than 1. It should be obvious that, along any given ellipse, the one for which the magnitude of the reflection coefficient is greatest is the one on the imaginary axis, since as we move along the ellipse towards that point, the distance to - Z_0 decreases and the distance from Z_0 increases (remember, their sum is constant along the ellipse). So on any *one* ellipse, the largest reflection coefficient magnitude occurs where the ellipse meets the imaginary axis, and thus Z_L has real part 0 and imaginary part of opposite sign to that of Z_0. (4) Start with Z_0 real, and slowly rotate Z_0 into either the first or fourth quadrant, but not more than 45 degrees in either direction, keeping the same magnitude while you rotate. The segments joining Z_0 and - Z_0, their perpendicular bisectors, and the various ellipses will all simultaneously rotate. It's now obvious that for ellipses of any fixed size, the one producing the largest magnitude for the reflection coefficient will occur when Z_0 is at + 45 degrees or - 45 degrees. So if we want to maximize the reflection coefficient magnitude, we can restrict attention to those two cases. The - 45 degree case (capacitive surge impedance) is the more familiar one, but the math is the same either way. The only question is, which one of the ellipses should we use, if we wish to maximize the magnitude of the reflection coefficient? (5) So now we're going to assume Z_0 = k(1 - j), and thus - Z_0 = k(-1 + j), while Z_L = ktj. It's clear that the factor k is going to cancel out when calculating the reflection coefficient, so I will henceforth ignore it (i.e., normalize it to k = 1 by appropriate choice of units). If you are of a geometrical turn of mind, you can produce a geometrical argument showing that the best one can do is to make sure the ellipse meets the imaginary axis at the same distance from the origin as the two foci, i.e. at j*sqrt(2). If you are of an algebraic turn of mind, you can make an algebraic argument involving completing the square to demonstrate the same thing. If you insist on using calculus, it's now just one variable calculus, not multivariable calculus, since the only independent variable is t, which will turn out to be sqrt(2) at the maximum. (Hint: don't look at the ratio of distances; look at the square of that ratio, so as to get rid of all those square roots.) (6) Once all that is done, it's just a bit of algebra to show that when t = sqrt(2) then the magnitude of the reflection coefficient is 1 + sqrt(2). That's the best (worst?) you can do. And if you can find some coax whose surge impedance angle is - 45 degrees, you can indeed do it. All the above was done from first principles. I am not fortunate enough to own a copy of Chipman, though I wish I were, but if this is what he says, then I am in full agreement with him. David, ex-W8EZE, willing to part with some of my pension money for a copy of Chipman if you know where one can be found (Powell's doesn't have any copies) -- David or Jo Anne Ryeburn To send e-mail, remove the letter "z" from this address. |
David or Jo Anne Ryeburn wrote:
David, ex-W8EZE, willing to part with some of my pension money for a copy of Chipman if you know where one can be found (Powell's doesn't have any copies) When looking for a copy yesterday, it seemed incredible that none of the major US dealers had even one for sale. However, copies might also have been catalogued under "Schaum" as the author, the book being one of the Schaum's Outline series. I only discovered this possibility when it was too late to do anything about it, so it may be worth a second look at the US dealers. -- 73 from Ian G3SEK 'In Practice' columnist for RadCom (RSGB) Editor, 'The VHF/UHF DX Book' http://www.ifwtech.co.uk/g3sek |
On Fri, 29 Aug 2003 08:16:35 +0100, "Ian White, G3SEK"
wrote: David or Jo Anne Ryeburn wrote: David, ex-W8EZE, willing to part with some of my pension money for a copy of Chipman if you know where one can be found (Powell's doesn't have any copies) When looking for a copy yesterday, it seemed incredible that none of the major US dealers had even one for sale. Hi All, One very reliable source gave me two other sources with multiple copies to choose from: http://www.bookfinder.com/search/?ac...0558_2:268:527 73's Richard Clark, KB7QHC |
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On Fri, 29 Aug 2003 17:28:24 -0400, "Tom Coates"
wrote: Search Amazon.com and www.addall.com for ISBN 0070107475. They're my favorite book-finding tools. Tom, N3IJ Hi Tom, Thanks for the new lead to a book site. 73's Richard Clark, KB7QHC |
Group:
Another very good and inexpensive reference on waves that might be of interest to this group is: William C. Elmore and Mark A Heald, "Physics of Waves", first published in 1969, but most recently published in paperback by Dover Publications, New York, 1985 and generally available in reprints even today for around US$17.00. A real bargain. ISBN: 0-486-64926-1 LCCN: 85-10419. Be aware, since this is a Physics book it is loaded with gratuitous partial differential equations. This book is interesting because it covers the *whole* field of waves, not just electromagnetic waves. Electromagnetic waves are particularly simple when compared to general wave phenomena, since em waves are transverse only. This book covers waves on strings and membranes, waves in and on fluids, waves in compressible media such as the earth [the seismic wavefield]etc... and so covers many of the "analogies" that posters to this news group like to draw upon, often drawing false conclusions. Are ocean waves at the beach *really* analogous to em waves? This book will explain why not. The earth supports both transverse and compressive-dillutive waves as well as surface seismic waves. Are compressive-dillutive waves different either qualitatively or quantitatively from electromagnetic transverse waves, this book answers the question. etc, etc... Expand your wave horizons beyond mere em waves, if you are deeply interested in waves, this book is more than worth the price at US$17.00. It's available from Amazon. -- Peter K1PO Indialantic By-the-Sea, FL "David or Jo Anne Ryeburn" wrote in message ... I've enjoyed reading this and related threads. Some comments have been made about using calculus. Though I spent a significant portion of my pre-retirement life attempting to teach that subject to undergraduates, I don't believe in using calculus whenever simple geometry and/or algebra makes it unnecessary. A proof that avoids calculus can be meaningful for those who don't know calculus, or who haven't used it for a while. With that said, I have a few comments to make about some of the assertions I have read here recently, some of which have appeared without explicit proof. (1) The surge impedance of a (lossy) transmission line cannot have an angle more than 45 degrees away from the real axis. This is true. Z_0 = sqrt((R + jwL)/(G + jwC)) (here I am using "w" instead of omega). Both the numerator and denominator lie in the first quadrant, so their quotient lies in the right half plane (angles subtract when one divides), and the square root of that has an angle lying between - 45 degrees and + 45 degrees. (The branch of the square root with positive real part has to be taken; if you can find coax whose surge impedance has a negative real component, I'll pay you good money for it.) [Since this angle lies between - 45 degrees and + 45 degrees, peculiar consequences deduced from calculations involving surge impedances such as 50 - j200 can be ignored.] (2) There is a nice geometrical interpretation for the reflection coefficient, or rather for its magnitude. Since the coefficient is (Z_L - Z_0)/(Z_L + Z_0), its magnitude expresses how much further Z_L is from the surge impedance Z_0 than it is from the negative, - Z_0, of the surge impedance. If Z_L is equidistant from Z_0 and - Z_0, then the magnitude of the reflection coefficient is 1. If Z_L is closer in the complex plane to Z_0 than it is to - Z_0, then the magnitude of the reflection coefficient is less than 1. If Z_L is closer to - Z_0 than to Z_0, then the reflection coefficient's magnitude exceeds 1. Now plot the points Z_0 and - Z_0 and draw the perpendicular bisector of the segment joining them. If Z_L is on that perpendicular bisector, the magnitude of the reflection coefficient is 1; if it is on Z_0's side of the bisector, the magnitude is less than 1; if it is on - Z_0's side, the magnitude exceeds 1. Of course Z_L has to stay in the right half plane; if it didn't have to do this, you could take Z_L very close to - Z_0 and get enormous reflection coefficient magnitudes. (3) Consider an ellipse having Z_0 and - Z_0 as its foci. There are infinitely many such ellipses, including a degenerate one (just the segment between the "foci"). All these different ellipses fill up the complex plane, and no point in the plane is on more than one of them. On any one such ellipse, the sum of the distances from a point on the ellipse to the two foci Z_0 and - Z_0 is constant (definition of an ellipse), the value of that constant depending upon which ellipse it is but the constant has to be at least as large as the interfocal distance. We should ignore points on the ellipse that are in the left half plane. A portion of the ellipse will be in the same quadrant as Z_0, and a portion will be in the quadrant that contains the conjugate of Z_0. (Remember we are ignoring the points in the left half plane.) All of the points on the ellipse that are in Z_0's quadrant are closer to Z_0 than to - Z_0, so they'll give reflection coefficients with magnitude less than 1. So will those of the points on the ellipse in the other quadrant under consideration that are between the real axis and the perpendicular bisector. But those that are between the perpendicular bisector and the imaginary axis will be closer to - Z_0 than to Z_0 and thus will yield reflection coefficients with magnitudes greater than 1. It should be obvious that, along any given ellipse, the one for which the magnitude of the reflection coefficient is greatest is the one on the imaginary axis, since as we move along the ellipse towards that point, the distance to - Z_0 decreases and the distance from Z_0 increases (remember, their sum is constant along the ellipse). So on any *one* ellipse, the largest reflection coefficient magnitude occurs where the ellipse meets the imaginary axis, and thus Z_L has real part 0 and imaginary part of opposite sign to that of Z_0. (4) Start with Z_0 real, and slowly rotate Z_0 into either the first or fourth quadrant, but not more than 45 degrees in either direction, keeping the same magnitude while you rotate. The segments joining Z_0 and - Z_0, their perpendicular bisectors, and the various ellipses will all simultaneously rotate. It's now obvious that for ellipses of any fixed size, the one producing the largest magnitude for the reflection coefficient will occur when Z_0 is at + 45 degrees or - 45 degrees. So if we want to maximize the reflection coefficient magnitude, we can restrict attention to those two cases. The - 45 degree case (capacitive surge impedance) is the more familiar one, but the math is the same either way. The only question is, which one of the ellipses should we use, if we wish to maximize the magnitude of the reflection coefficient? (5) So now we're going to assume Z_0 = k(1 - j), and thus - Z_0 = k(-1 + j), while Z_L = ktj. It's clear that the factor k is going to cancel out when calculating the reflection coefficient, so I will henceforth ignore it (i.e., normalize it to k = 1 by appropriate choice of units). If you are of a geometrical turn of mind, you can produce a geometrical argument showing that the best one can do is to make sure the ellipse meets the imaginary axis at the same distance from the origin as the two foci, i.e. at j*sqrt(2). If you are of an algebraic turn of mind, you can make an algebraic argument involving completing the square to demonstrate the same thing. If you insist on using calculus, it's now just one variable calculus, not multivariable calculus, since the only independent variable is t, which will turn out to be sqrt(2) at the maximum. (Hint: don't look at the ratio of distances; look at the square of that ratio, so as to get rid of all those square roots.) (6) Once all that is done, it's just a bit of algebra to show that when t = sqrt(2) then the magnitude of the reflection coefficient is 1 + sqrt(2). That's the best (worst?) you can do. And if you can find some coax whose surge impedance angle is - 45 degrees, you can indeed do it. All the above was done from first principles. I am not fortunate enough to own a copy of Chipman, though I wish I were, but if this is what he says, then I am in full agreement with him. David, ex-W8EZE, willing to part with some of my pension money for a copy of Chipman if you know where one can be found (Powell's doesn't have any copies) -- David or Jo Anne Ryeburn To send e-mail, remove the letter "z" from this address. |
Roy:
[snip] I thought this was only true for waves moving through a lossless medium or in a lossless transmission line that supports TEM waves. Either the electric or magnetic field isn't transverse in a hollow waveguide, and either or both can be non-transverse in a lossy medium. Or am I mistaken about this? Roy Lewallen, W7EL [snip] Compressive-dillutive waves occur only in media that is compressible, like the earth or the air, or springs, etc... With compressive-dillutive waves the "vibrations" occur in the effective density of the medium. Electromagnetic waves propagate with transversal vibrations of the E and H fields, viz. Side to side vibrations, not shortening and lenghtening vibrations. If an electromagnetic wave is supported in the Transverse Electromagnetic or TEM mode then [theoretically] the E and H fields are at right angles to each other and to the direction of propagation, in the terminology of TEM, the term transverse referes to the orientation of the wave with respect to it's guides, and not to it's vibration mode in the longitudinal - transverse sense. Visualize a long "slinky" coil attached to the wall. Shake one end up and down to create a transverse wave in which the slinky moves up and down. Push and pull on the end to produce compression and dilution to cause longitudinal waves in which the slinky does not move up and down but in which the distance between turns moves back and forth. This slinky analogy sort of illustrates the differences. Meanwhile in electromagnetic wave phenomena you have as well as the most common TEM mode which is only transverse vibrations, also there exists a plethora of TM and/or TE modes, or even in the near field, where the fields may not be at right angles to each other or to the direction of propagation, but the vibrations are still talways transverse, i.e. not compressive-dilutive. Back in the mid-1800's after Maxwell produced his celebrated equations and Heaviside improved them by expressing them in vector form most scientists of the time noted that Maxwell's formulation provided no explicit form for a medium for the electromagnetic waves to propagate in, and they also noted that there were only transverse and not longitudinal [compressive-dilutive] vibrations supported by his equations. Several eminent scientists of the day felt that this left openings for several more discoveries and so... Then ensued for several decades a search for the "ether". The "ether" was supposed to be the media which supported the electromagnetic waves. During that period several of the eminent scientists of the time proposed that the "ether" once it was found might actually be compressible and they proposed that Maxwell and Heaviside had left out of their formulations the possibility of compressive-dilutive or longitudinal vibrations. Several scientists of the time actually formulated equations which supported compressive-dilutive em waves and actually conconcted and, to no avail, actually conducted experiments to try to find out if such compressive-dilutive vibrations actually occured with electromagnetic phenomena. As we all know, eventually the existence of the "ether" was discredited, mainly by the Michelson-Morley experiments, and today we all know that electromagnetic waves do not have a media or "ether" to support their propagation and vibrations. Electromagnetic waves propagate just fine in a complete vacuum, and a vacuum is incompressible, and so the search for compressive-dilutive vibrations of electromagnetic waves became moot and a search for experimental evidence of them was abandoned by all who were interested. One can add terms to the Maxwell-Heaviside equations to support compressive waves, and this has been done by several theoretical physicists, but there is no sense doing so since none have ever been discovered! The book, I referred to above, "Physics of Waves" gives all the details of the wave equation for media that supports compressive waves. An important such field is the field of seismology. Indeed the field of siesmology studies waves that vibrate in all modes, transversally and longitudinally, as well as surface waves. Seismic waves are processed regularly with beam forming arrays of seismometers and processed by tomographic techniques to image the earth in all wave modes. Seismology is a facinating field and seismologists are generally the most sophisticated of all wave mechaics! A good modern book on the seismic wavefield is: B. L. N. Kennett, "The Siesmic Wavefield", Cambridge University Press, New York, NY, 2001. ISBN: 0-521-00663-5. But be aware it is full of gratuitous partial differential equations and tensor analysis. The stress-strain variables of compressible-dillutive media are expressed as tensors and the partial differential equations are cast in tensor form. All this to say that electromagnetic wave phenomena are a particularly simple form of wave phenomena when compared to the most complicated types. -- Peter K1PO Indialantic By-the-Sea, FL. |
Thanks for the most interesting discussion of slinkys, "ether", and
seismology. But I'm a little vague on what you mean by "vibrations". You're describing a field whose orientation isn't necessarily at a right angle (transverse) to the direction of propagation (as in a TE or TM mode wave), yet whose "vibrations" are nevertheless at a right angle to the direction of propagation. So the "vibrations" are in a different direction than the field. I'd like to learn more about this phenomenon, but I can't find "vibrations" in the indexes of any of my electromagnetics texts. Do they have another name? Roy Lewallen, W7EL Peter O. Brackett wrote: . . . Visualize a long "slinky" coil attached to the wall. Shake one end up and down to create a transverse wave in which the slinky moves up and down. Push and pull on the end to produce compression and dilution to cause longitudinal waves in which the slinky does not move up and down but in which the distance between turns moves back and forth. This slinky analogy sort of illustrates the differences. Meanwhile in electromagnetic wave phenomena you have as well as the most common TEM mode which is only transverse vibrations, also there exists a plethora of TM and/or TE modes, or even in the near field, where the fields may not be at right angles to each other or to the direction of propagation, but the vibrations are still talways transverse, i.e. not compressive-dilutive. . . . |
Peter O. Brackett wrote:
Electromagnetic waves propagate with transversal vibrations of the E and H fields, ... Heh, heh, not wearing the particle physics hat today, Peter? :-) -- 73, Cecil http://www.qsl.net/w5dxp -----= Posted via Newsfeeds.Com, Uncensored Usenet News =----- http://www.newsfeeds.com - The #1 Newsgroup Service in the World! -----== Over 100,000 Newsgroups - 19 Different Servers! =----- |
Thanks for the most interesting discussion of slinkys, "ether", and seismology. But I'm a little vague on what you mean by "vibrations". You're describing a field whose orientation isn't necessarily at a right angle (transverse) to the direction of propagation (as in a TE or TM mode wave), yet whose "vibrations" are nevertheless at a right angle to the direction of propagation. So the "vibrations" are in a different direction than the field. I'd like to learn more about this phenomenon, but I can't find "vibrations" in the indexes of any of my electromagnetics texts. Do they have another name? Roy Lewallen, W7EL Yes, "vibes." I think Peter is regressing back to the '60's. 73, Tom Donaly, KA6RUH |
Roy:
[snip] "Roy Lewallen" wrote in message ... Thanks for the most interesting discussion of slinkys, "ether", and seismology. But I'm a little vague on what you mean by "vibrations". You're describing a field whose orientation isn't necessarily at a right angle (transverse) to the direction of propagation (as in a TE or TM mode wave), yet whose "vibrations" are nevertheless at a right angle to the direction of propagation. So the "vibrations" are in a different direction than the field. I'd like to learn more about this phenomenon, but I can't find "vibrations" in the indexes of any of my electromagnetics texts. Do they have another name? Roy Lewallen, W7EL [snip] Vibrations = oscillations An instance where compressional-dilative waves might occur in electromagnetic propagation and where those compressional vibrations terms could be added to the Maxwell-Heaviside equations might be that of electromagnetic propagation through "light ion" plasmas [ionized gases] where the ions could physically respond essentially instantaeously to the passing waves and the distance between ions and hence the media properties becomes a function of the electromagnetic fields. The effective mu and epsilon of the media changing instantaneously in response to the propagating fields, in turn changing the waves, etc... just as for compression acoustic wave propagating in a compressible gas. This effect is probably infinitesimal for "heavy ion" plasmas and might be perceptable for "light ion" plasmas. I wonder if any readers of this NG have any experience with propagation in plasmas and can share with us if they use compression-dilutive terms to augment the Maxwell-Heaviside equations in the analysis. I presume that the NEC code that you use in EZNEC to integrate the Maxwellian equations does not support plasma propagation analysis. Perhaps someone knows of a version of NEC that does. I'd guess that folks at Lawrence Livermore and at NASA are interested in such problems. I'd be curious to know if they use augmented versions of Maxwell-Heaviside equations. Another, arcane, far fetched, and impractical example of compressional-dillutive vibrations in em waves that I can think of could be imagined as a system wherein em waves travel in a waveguide system where the dimensions of the system [walls of the waveguide] are such that they can move in and out instantaneously in response to the passing waves thus alternately confining and expanding the dimensions of the guide relative to the wavelength of the passing waves, it might be imagined that such action could induce a wave shortening and lengthening effect on the passing waves which is what compression-dillution waves are. Thoughts, comments? -- Peter K1PO Indialantic By-the-Sea, FL. |
Tom:
[snip] Yes, "vibes." I think Peter is regressing back to the '60's. 73, Tom Donaly, KA6RUH [snip] Hmmm.... I remember Angela fondly, and how we used to dance to "I just wanna hold your hand..." Now just where did I put those old Beatles albums? -- Peter K1PO Indialantic By-the-Sea, FL |
Peter O. Brackett wrote:
Say did you order your copy of "Entanglement" yet? I'm going to take a look at it first at the Texas A&M library hopefully next week. What is Cecil's wavelength and can I get on it? Want me to grid dip myself? -- 73, Cecil http://www.qsl.net/w5dxp -----= Posted via Newsfeeds.Com, Uncensored Usenet News =----- http://www.newsfeeds.com - The #1 Newsgroup Service in the World! -----== Over 100,000 Newsgroups - 19 Different Servers! =----- |
Thanks for the rundown on "light ion" plasmas, plasma propagation, and
moving-wall waveguides. I only have one remaining question. Do the "vibrations" of electromagnetic waves you referred to in your previous post have another name? Roy Lewallen, W7EL Peter O. Brackett wrote: Roy: [snip] "Roy Lewallen" wrote in message ... Thanks for the most interesting discussion of slinkys, "ether", and seismology. But I'm a little vague on what you mean by "vibrations". You're describing a field whose orientation isn't necessarily at a right angle (transverse) to the direction of propagation (as in a TE or TM mode wave), yet whose "vibrations" are nevertheless at a right angle to the direction of propagation. So the "vibrations" are in a different direction than the field. I'd like to learn more about this phenomenon, but I can't find "vibrations" in the indexes of any of my electromagnetics texts. Do they have another name? Roy Lewallen, W7EL [snip] Vibrations = oscillations An instance where compressional-dilative waves might occur in electromagnetic propagation and where those compressional vibrations terms could be added to the Maxwell-Heaviside equations might be that of electromagnetic propagation through "light ion" plasmas [ionized gases] where the ions could physically respond essentially instantaeously to the passing waves and the distance between ions and hence the media properties becomes a function of the electromagnetic fields. The effective mu and epsilon of the media changing instantaneously in response to the propagating fields, in turn changing the waves, etc... just as for compression acoustic wave propagating in a compressible gas. This effect is probably infinitesimal for "heavy ion" plasmas and might be perceptable for "light ion" plasmas. I wonder if any readers of this NG have any experience with propagation in plasmas and can share with us if they use compression-dilutive terms to augment the Maxwell-Heaviside equations in the analysis. I presume that the NEC code that you use in EZNEC to integrate the Maxwellian equations does not support plasma propagation analysis. Perhaps someone knows of a version of NEC that does. I'd guess that folks at Lawrence Livermore and at NASA are interested in such problems. I'd be curious to know if they use augmented versions of Maxwell-Heaviside equations. Another, arcane, far fetched, and impractical example of compressional-dillutive vibrations in em waves that I can think of could be imagined as a system wherein em waves travel in a waveguide system where the dimensions of the system [walls of the waveguide] are such that they can move in and out instantaneously in response to the passing waves thus alternately confining and expanding the dimensions of the guide relative to the wavelength of the passing waves, it might be imagined that such action could induce a wave shortening and lengthening effect on the passing waves which is what compression-dillution waves are. Thoughts, comments? -- Peter K1PO Indialantic By-the-Sea, FL. |
Roy:
[snip] Do the "vibrations" of electromagnetic waves you referred to in your previous post have another name? Roy Lewallen, W7EL [snip] Oscillations perhpaps? I don't really understand your question... do you object to the term "vibrations"? What would you prefer, oscillations, or... It is well known by Physicists that lectromagnetic waves [at least in free space and isotropic media] are generally consist of only transverse vibrations,. this type of vibration is inherent in the formulation and solutions to the Maxwell-Heaviside equations. For examples of longitudinal or compressive vibratons for instance in a taught wire like a guitar string, transverse vibrations or oscillations are side to side, but longitudinal or compressional vibrations would be the very tiny vibrations in the length of the guitar string. In systems where longitudinal vibrations are supported, generally the velocity of propagation of longitudinal vibrations will not be the same as that of transverse vibrations. For a detailed explantation of compressional-dilutive or longitudinal waves in a variety of physical systems, cfr: William C. Elmore, and Mark A. Heald, "Physics of Waves", McGraw-Hill, New York, 1969. -- Peter K1PO Indialantic By-the-Sea, FL. |
No, I'm just trying to figure out how an EM wave can have E and/or H
fields whose directions aren't transverse to the direction of propagation (as in a lossy medium or in a hollow waveguide) can have another property of "vibrations" or "oscillations" that *are* always transverse. The properties of E and H fields that I'm familiar with include orientation of the field in space, and change in amplitude with time (which is what I'd normally call oscillations). And, of course, the orientation can change in time also, as in elliptically or circularly polarized waves. The property of "vibrations" or "oscillations" that have a direction different from the direction of the field is new to me. I notice that you're now qualifying your statement to free space and isotropic media. Does this perhaps leave open the possibility that waves in a lossy medium, or bounded within a hollow waveguide, could have "vibrations" that *aren't* transverse to the direction of propagation? My original question was in response to your statement that EM waves were always transverse, regardless of the medium. Do you perhaps have Krus' _Electromagnetics_, or electromagnetics texts by Holt, Johnk, Skilling, Magid, Magnusson, or Jordan & Balmain? If so, perhaps you could direct me to a section which addresses this. Roy Lewallen, W7EL Peter O. Brackett wrote: Roy: [snip] Do the "vibrations" of electromagnetic waves you referred to in your previous post have another name? Roy Lewallen, W7EL [snip] Oscillations perhpaps? I don't really understand your question... do you object to the term "vibrations"? What would you prefer, oscillations, or... It is well known by Physicists that lectromagnetic waves [at least in free space and isotropic media] are generally consist of only transverse vibrations,. this type of vibration is inherent in the formulation and solutions to the Maxwell-Heaviside equations. For examples of longitudinal or compressive vibratons for instance in a taught wire like a guitar string, transverse vibrations or oscillations are side to side, but longitudinal or compressional vibrations would be the very tiny vibrations in the length of the guitar string. In systems where longitudinal vibrations are supported, generally the velocity of propagation of longitudinal vibrations will not be the same as that of transverse vibrations. For a detailed explantation of compressional-dilutive or longitudinal waves in a variety of physical systems, cfr: William C. Elmore, and Mark A. Heald, "Physics of Waves", McGraw-Hill, New York, 1969. -- Peter K1PO Indialantic By-the-Sea, FL. |
Roy:
[snip] "Roy Lewallen" wrote in message ... The property of "vibrations" or "oscillations" that have a direction different from the direction of the field is new to me. [snip] That's fine, there are lots of things new to me as well. :-) These are called longitudinal or compressive-dilutive waves. Such vibrations do not [usually] occur with electromagnetic fields, and in the early days of em theory scientists did wonder if such were possible, as I have noted in prior postings. However there are numerous physical systems described by wave equations which do support both transverse and longitudinal vibrations of the constituient fields. I gave several examples in prior postings and you will find lots of such examples in "Physics of Waves". [snip] I notice that you're now qualifying your statement to free space and isotropic media. Does this perhaps leave open the possibility that waves in a lossy medium, or bounded within a hollow waveguide, could have "vibrations" that *aren't* transverse to the direction of propagation? [snip] The point is not that the waves have "vibrations" that are transverse to the direction of propagation. Of course, guided em waves have vibrations which are not perfectly perpendicular to the direction of propagation. The TEM mode would not exist and TEM waves would not propagate if there were not some potential driving the waves forward, meaning that both the E and H fields have some tiny component in the direction of propagation. All of this to establish that I do understand em wave propagation, and to say that... this has nothing whatsoever to do with longitudinal or compressive-dilutive vibrations. The fact that E and H fields "lean" slightly in the direction of propagation in a wave guide is not a compressive-dillutive effect on the fields. For longitudinal field vibrations to occur the wavelength of the propagating fields has to change as it propagates. This does not occur in "normal" em propagation. I conjecture that there may however be some exceptions to this, e.g. plasmas, etc... I just am not aware of them. Perhaps some other newsgroup reader/poster is more familiar with any possible longitudinal vibrations of em waves. [snip] My original question was in response to your statement that EM waves were always transverse, regardless of the medium. [snip] Roy, I believe that you may be reading too much into the word "transverse", it can be used in several contexts. Tansverse vibrations are not compressive vibrations. With compressive vibrations, the wavelength of the waves actually changess it propagates. While in transverse vibrations no such wavelength changes occur. In this usage the word "transverse" does not refer to directionality with respect to direction of propagation, but rather to the fact that the waves maintain their wavelegth during propagation. As a "real" example, some seismic waves [the so called "S-Waves" in the earth actually change their wavelength as they propagate.. [snip] Do you perhaps have Krus' _Electromagnetics_, or electromagnetics texts by Holt, Johnk, Skilling, Magid, Magnusson, or Jordan & Balmain? If so, perhaps you could direct me to a section which addresses this. [snip] Roy, yes indeed I have two editions of "Kraus" and I took a course from Keith Balmain using his first edition text when I was at U of T. And... I can tell you here and now that neither of those two august gentlemen address the issue of longitudinal vibrations anywhere in their texbooks! Simply because, as I have stated in other postings, Maxwell-Heaviside equations do not support longitudinal vibrations, and so why would a text on em waves even discuss such vibrations? The fact that Kraus and Balmain do not discuss such things does not surprise me, nor should it you, since electromagnetic wave propagation and the Maxwell-Heaviside equations are a particularly simple example of wave motion. Roy if you wish to deeply understand wave equations and wave motions and to understand the wider ramifications of wave motion, you just gotta read more widely in the "Waves" literature. Kraus and Balmain are very narrow in scope, being confined strictly to em waves! If they had attempted to include any "early" history of em research from around the middle of the 1800's then they would have outlined some of the early speculations by contemporaries of Maxwell, such as Kelvin, Heaviside and others as to the possibility that Maxwell might have left longitudinal terms that might have proved significant, of course they were found never to be needed. However even back in those times most Natural Philosophers [They were'nt called Physicists in those days] and Electricians like Heaviside were more widely schooled than today's Engineers and they knew and studied wave equations in their full glory... longitudinal vibrations included. These days however our electrical engineering education is far too narrrow and does not expose folks to the wider view of the world. Thus we often find em wave mechanics who don't understand longitudinal waves. Until I became involved in underwater acoustics and seismic propagation problems and saw the wave equations in their full glory, I too had a narrow view of wave mechanics. I don't know if Kraus or Balmain ever encountered the "full" wave equations, but in any case their texts are directed at em specialists and so their narrow view is not surprising. As I posted before, if you are interested in such things, check out: Elmore and Heald, "Physics of Waves" and say, Kennett's, "The Seismic Wavefield" among others to help you to broaden your horizons on these issues of longitudinal waves. -- Peter K1PO Indialantic By-the-Sea, FL. |
Dear Professor David or Jo Anne Ryeburn,
Finally, I accomplished the study of this most interesting article... But, with your permission, I can not resist to notice that the key-point of the surprising, at least to me, introduction of an ellipse at step (3), it looks somehow artificial and in some way opposite to the intentions of the introduction: | ... | don't believe in using calculus | whenever simple geometry and/or algebra | makes it unnecessary. | A proof that avoids calculus can be meaningful for | those who don't know calculus, | or who haven't used it for a while | ... In all other respects and as far as I could say something more, then it is, at least for me, a perfect argument, indeed! Sincerely yours, pez SV7BAX TheDAG |
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