Therein lies the problem - there is no definition of "cutoff" for anything but an infinite exponential horn. All other usages of the term thus become prone to individual interpretation leading to the kind of semantic discussions that we are having here.
But if we choose to define "cutoff" in the same sense as a filter, the 1/2 power points, and we do this for the infinite horn case, then my example above is completely correct - the long and short conical horns having the same "cutoff". But if one is going to take the "ripples" cause by mouth reflections into consideration then there is no answer because each length and situation is different - it depends on where the resonance lie relative to the infinite devices "cutoff" - and then there is the flared mouth case which adds even more complication to the problem.
A rather complex answer, I know, but so much of the endless bickering that goes on here is because of loose definitions of terms, hence it is critical to lay out what we mean when we say "cutoff". As I said, the only definition for horns that one can find in the literature is for the frequency below which no sound is transmitted down an exponential horn. But, since this phenomena is a direct result of "horn equation" approximation errors, and does not really occur in reality, it becomes anybody's guess what "cutoff' means in the real world.
This is precisely why I said that I abhor the term. Its like "waveguide" versus "horn" - the terms have become so abused as to loose all meaning.
But if we choose to define "cutoff" in the same sense as a filter, the 1/2 power points, and we do this for the infinite horn case, then my example above is completely correct - the long and short conical horns having the same "cutoff". But if one is going to take the "ripples" cause by mouth reflections into consideration then there is no answer because each length and situation is different - it depends on where the resonance lie relative to the infinite devices "cutoff" - and then there is the flared mouth case which adds even more complication to the problem.
A rather complex answer, I know, but so much of the endless bickering that goes on here is because of loose definitions of terms, hence it is critical to lay out what we mean when we say "cutoff". As I said, the only definition for horns that one can find in the literature is for the frequency below which no sound is transmitted down an exponential horn. But, since this phenomena is a direct result of "horn equation" approximation errors, and does not really occur in reality, it becomes anybody's guess what "cutoff' means in the real world.
This is precisely why I said that I abhor the term. Its like "waveguide" versus "horn" - the terms have become so abused as to loose all meaning.
Thanks Earl, I understand your answer. As I have not ever seen a precise definition of "Horn Cutoff" so I thought I might have missed something. Looks like there isn't one, eh?
When I'm working on crossovers I define cutoff as the frequency below which on axis response starts dropping rapidly and H2 and H3 take a turn sharply upward. It's usually easy to see in measurements. I try to stay an octave above that point with the crossover.
I understand how other people might define cutoff differently, perhaps related to pattern control or something else.
When I'm working on crossovers I define cutoff as the frequency below which on axis response starts dropping rapidly and H2 and H3 take a turn sharply upward. It's usually easy to see in measurements. I try to stay an octave above that point with the crossover.
I understand how other people might define cutoff differently, perhaps related to pattern control or something else.
Huh!
"As far as waveguides versus horns, you guys need to read my explanation on my website. You may not like it, but that is beside the point. Its how I see the issues and this thread is titled "Geddes on Waveguides"".
This is a public space called the Internet in which this thread is placed. Thus, the thread is not a refuge from the facts nor the opinions of others. You may of course, call it anything you want, but that does not change what it is whether it be space on the internet or an acoustic horn. As usual the primary function is being confused with those that are secondary. Most engineers and physicists know the principal differences between dispersive radiators, collective receivers, and transmission devices. Apparently for your marketing reasons, we are expected to accept the delusion that some horns do not have a secondary "waveguide" function. NONSENSE!
WHG
"As far as waveguides versus horns, you guys need to read my explanation on my website. You may not like it, but that is beside the point. Its how I see the issues and this thread is titled "Geddes on Waveguides"".
This is a public space called the Internet in which this thread is placed. Thus, the thread is not a refuge from the facts nor the opinions of others. You may of course, call it anything you want, but that does not change what it is whether it be space on the internet or an acoustic horn. As usual the primary function is being confused with those that are secondary. Most engineers and physicists know the principal differences between dispersive radiators, collective receivers, and transmission devices. Apparently for your marketing reasons, we are expected to accept the delusion that some horns do not have a secondary "waveguide" function. NONSENSE!
WHG
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Earl,
You are right, my statement "A larger diameter round conical horn with more mouth area has a lower cutoff frequency than a smaller horn" was wrong.
I finally went to do a simulation in Hornresp to prove that a larger diameter conical horn would have a lower Fc than a smaller diameter horn with the same side wall angle.
My first attempt resulted in approximately 1/3 octave lower Fc for doubling the length, but it was due to the horn angle being slightly less (exactly as you wrote), once the angle was corrected to be the same the FC remained the same.
The results seem counter intuitive, thank you for pointing out my error enough times that I finally did the comparison.
That said, my 90 x 40 degree 10.75" x 5" horn does go lower than a 5" x 5" 90 x 90 degree waveguide, but it is because of the narrow vertical angle (which does result in pattern flip), not mouth size.
At any rate, the lower Fc and greater on axis HF response of the wider horn worked well for my application, again I thank you for "sticking to your guns" so I finally did the work to understood the reason why it went lower.
Art
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Art
And thank you for acknowledging that every once in a while I actually do know what I am talking about.
WHG
I have no idea what you are talking about, but posting results from a theory that has been completely debunked does not make much of an impression.
Pano
The only recognized definition is that frequency where the argument of the solution to wave propagation (a complex exponential) goes to zero. Above that frequency the wavenumber is real and sound propagates. Below that frequency the wavenumber is imaginary and the wave becomes "evanescent" - a damped exponential with distance. This phenomena occurs for the higher modes and this frequency is called "cutoff" there as well. In the (incorrect) solution to an infinite exponential horn the same thing occurs, but in an exact solution to the waveguide problem for the fundamental mode there is never an evanescent wave and no cutoff in the precise definition of the term.
And thank you for acknowledging that every once in a while I actually do know what I am talking about.
WHG
I have no idea what you are talking about, but posting results from a theory that has been completely debunked does not make much of an impression.
Pano
The only recognized definition is that frequency where the argument of the solution to wave propagation (a complex exponential) goes to zero. Above that frequency the wavenumber is real and sound propagates. Below that frequency the wavenumber is imaginary and the wave becomes "evanescent" - a damped exponential with distance. This phenomena occurs for the higher modes and this frequency is called "cutoff" there as well. In the (incorrect) solution to an infinite exponential horn the same thing occurs, but in an exact solution to the waveguide problem for the fundamental mode there is never an evanescent wave and no cutoff in the precise definition of the term.
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More
"I have no idea what you are talking about, but posting results from a theory that has been completely debunked does not make much of an impression."
Limitations imposed by the simplifying assumption of “Plane Wave Propagation” within a horn were well known by Olson at the time he created the graph. Thus, his work shown here does not require any "debunking" by you. The graphs show the acoustical differences between various profiles under these assumptions for horns of infinite extent. It is also known, that there will be undulating departures from them for horns of finite extent as well as the emission of energy below the hypothetical [fc]'s shown. These facts should make further validation here unnecessary.
WHG
"I have no idea what you are talking about, but posting results from a theory that has been completely debunked does not make much of an impression."
Limitations imposed by the simplifying assumption of “Plane Wave Propagation” within a horn were well known by Olson at the time he created the graph. Thus, his work shown here does not require any "debunking" by you. The graphs show the acoustical differences between various profiles under these assumptions for horns of infinite extent. It is also known, that there will be undulating departures from them for horns of finite extent as well as the emission of energy below the hypothetical [fc]'s shown. These facts should make further validation here unnecessary.
WHG
;-)
Then show a graph you consider to be a correct comparison of the various horn profiles to support your assertion. Any model of reality will exhibit departures from it. That fact does not make the model "wrong".
WHG
Then show a graph you consider to be a correct comparison of the various horn profiles to support your assertion. Any model of reality will exhibit departures from it. That fact does not make the model "wrong".
WHG
I don't think that you understand the situation. None of the horn profiles shown are within the assumptions of plane wave propagation so none of them are correct. There is no way to show "a correct comparison of the various horn profiles" because there is no way to analyze them (short of numerical methods which is far too much work to do just to make a point).
There are good models and bad models - the Horn equation is a bad model because no contour can meet its assumptions. This is all well documented in my writings - maybe you should read them.
I do know this much: Since the OS waveguide is exact - there are no assumptions made - one can use its results to determine the errors that we should expect from Webster's approach. I did exactly that. I put the OS contour into the Horn equation and calculated the results. It was not an easy equation to solve, unlike the normal horn equation contours (which is obviously why they were selected in the first place). The results showed that the Horn equation is least accurate right at what one would call cutoff. At VHF the results converge - as they should - and at LF they converge - both going to zero (Dah!) but right at cutoff the two differed quite a bit - the wave equation approach being exact and accurate and the Horn equation results being less than attractive.
Maybe I'll post them if I can find them.
There are good models and bad models - the Horn equation is a bad model because no contour can meet its assumptions. This is all well documented in my writings - maybe you should read them.
I do know this much: Since the OS waveguide is exact - there are no assumptions made - one can use its results to determine the errors that we should expect from Webster's approach. I did exactly that. I put the OS contour into the Horn equation and calculated the results. It was not an easy equation to solve, unlike the normal horn equation contours (which is obviously why they were selected in the first place). The results showed that the Horn equation is least accurate right at what one would call cutoff. At VHF the results converge - as they should - and at LF they converge - both going to zero (Dah!) but right at cutoff the two differed quite a bit - the wave equation approach being exact and accurate and the Horn equation results being less than attractive.
Maybe I'll post them if I can find them.
Hi Earl,
When solving the horn equation for the OS contour, isn't the "cutoff" a function of x? Salmon comments on this in his papers: "Thus at the throat, where alpha [normalized axial length] =0, the region of transmission lies above mu=1. [mu = normalized frequency] However, there is no sharp cut-off, since the coefficient is a function of alpha." He then uses numerical integration of the horn equation from a point well removed from alpha=0 back to the throat to "obtain a more quantitative picture of the behavior" of the horn. Did you take this variation of "cutoff" with x into account in your calculations?
Regards,
Bjørn
When solving the horn equation for the OS contour, isn't the "cutoff" a function of x? Salmon comments on this in his papers: "Thus at the throat, where alpha [normalized axial length] =0, the region of transmission lies above mu=1. [mu = normalized frequency] However, there is no sharp cut-off, since the coefficient is a function of alpha." He then uses numerical integration of the horn equation from a point well removed from alpha=0 back to the throat to "obtain a more quantitative picture of the behavior" of the horn. Did you take this variation of "cutoff" with x into account in your calculations?
Regards,
Bjørn
Bjorn
You lost me.
"There is no sharp cut-off" but that doesn't mean that it is "a function of x" (whatever that means, I don't understand). We are talking about input impedance, correct? There is only one "x" - the throat location. If you pick a different "x" then you have a different device and, of course, the input impedance is going to be different.
You lost me.
"There is no sharp cut-off" but that doesn't mean that it is "a function of x" (whatever that means, I don't understand). We are talking about input impedance, correct? There is only one "x" - the throat location. If you pick a different "x" then you have a different device and, of course, the input impedance is going to be different.
Who's Model? How Bad is It?
“Freehafer’s exact theory for the hyperbolic horn” (Read Geddes’ OS Waveguide) was known to Salmon back in 1945 [R1]. The latter third of his treatise entitled “Generalized Plane Wave Horn Theory” [R2] was devoted to address these differences and show just how nominal they are.
Note: The referenced ASA Journal article has been redacted to show only the information relevant to the issue at hand here. This action is necessary in order to comply with the fair use provisions of U.S. copyright law.
WHG
I don't think that you understand the situation. None of the horn profiles shown are within the assumptions of plane wave propagation so none of them are correct. There is no way to show "a correct comparison of the various horn profiles" because there is no way to analyze them (short of numerical methods which is far too much work to do just to make a point).
There are good models and bad models - the Horn equation is a bad model because no contour can meet its assumptions. This is all well documented in my writings - maybe you should read them.
I do know this much: Since the OS waveguide is exact - there are no assumptions made - one can use its results to determine the errors that we should expect from Webster's approach. I did exactly that. I put the OS contour into the Horn equation and calculated the results. It was not an easy equation to solve, unlike the normal horn equation contours (which is obviously why they were selected in the first place). The results showed that the Horn equation is least accurate right at what one would call cutoff. At VHF the results converge - as they should - and at LF they converge - both going to zero (Dah!) but right at cutoff the two differed quite a bit - the wave equation approach being exact and accurate and the Horn equation results being less than attractive.
Maybe I'll post them if I can find them.
“Freehafer’s exact theory for the hyperbolic horn” (Read Geddes’ OS Waveguide) was known to Salmon back in 1945 [R1]. The latter third of his treatise entitled “Generalized Plane Wave Horn Theory” [R2] was devoted to address these differences and show just how nominal they are.
Note: The referenced ASA Journal article has been redacted to show only the information relevant to the issue at hand here. This action is necessary in order to comply with the fair use provisions of U.S. copyright law.
WHG
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At any point in the horn, the impedance is given as p/U, and as such, you could plot the ratio of p/U as function of the distance from the throat. In an exponential horn, this ratio, apart from a dependence on area, is constant, so the cutoff frequency is the same throughout the horn. If you had a conical horn approximated by a series of very short exponential horns, you would find that each of these short segments would have different cutoff frequencies (in terms of the infinite horn with the same flare rate), i.e. the cutoff would vary as a function of x. To just take the first of these segments, the one nearest the throat, and say that it defined a sharp cutoff in a conical horn would be nonsense. Each segment transforms the impedance seen at its mouth end, and cannot be viewed in isolation.
So yes, we are looking at the throat location, but the impedance at the throat is dependent on how the impedance varies further down the horn. The OS approaches a conical horn, which we know does not have a cutoff. If the horn is cut some distance from x=0, it will have nearly the same throat impedance as a conical horn. A short extra segment near the throat cannot create a sharp cutoff like what is found in the exponential horn. The sharp cutoff phenomenon appears in the exponential horn because the cutoff frequency does not depend on x.
Note: I'm discussing this based on the horn equation, which I'm very aware is inaccurate. I'm also only considering infinite horns. Real life will be different, and you will probably say this discussion is meaningless. But I think it is important, when comparing two solutions, to take the details into account, if we are to say anything about how good one is compared to the other. It's a different thing if we are to decide which one to use. But I believe your intent with the calculations you did was to show how well (or not) the horn equation holds up to an exact solution. I'm not here to defend the horn equation, I just want to give it a fair trial before the hanging
Regards,
Bjørn
Thanks for that paper, I was not familiar with it. But if you call the differences "nominal" then we have different views of what accuracy is. To me his comparison is exactly the same results that I got. No need to dig up mine now.
Maybe not "meaningless" but I don't see any point to it. I did the comparison exactly like Salmon did (as shown above) and got pretty much the same results. The errors in the horn equation are substantial and are greatest right at "cutoff". That's what I said and Salmon agrees. I'm satisfied with that.
I'm not here to defend the horn equation, I just want to give it a fair trial before the hanging
Regards,
Bjørn
Bjorn
One last point.
If you are going to continue to use the Horn equation then you are going to have to defend it. Thats the way of the world.
I think that you will find that defending it will be very difficult, because it is wrong, we all know its wrong and continuing to use it is wrong. Surely you must see that.
Its time to get on the right side of this debate.
Earl,
The term oblate spheroid is a nice mathematical term but doesn't really tell you much. A circle in effect can fall under this term as it is a subset of an ellipse with equal minor and major axis. All you are really saying with that terminology is that the shape is elliptical. That does not say anything about the transition from a round initial start point and the subsequent rate of change. You must use some rate of change from the initial x axis at the 0 point and the distance along the X axis. I could make an oblate spheroid that follows any expansion rate from conic to exponential along the X axis. So you are in effect not really defining the shape beyond its simplest basis. How does any of this define the length of the waveguide and the rate of expansion of the waveform?
The term oblate spheroid is a nice mathematical term but doesn't really tell you much. A circle in effect can fall under this term as it is a subset of an ellipse with equal minor and major axis. All you are really saying with that terminology is that the shape is elliptical. That does not say anything about the transition from a round initial start point and the subsequent rate of change. You must use some rate of change from the initial x axis at the 0 point and the distance along the X axis. I could make an oblate spheroid that follows any expansion rate from conic to exponential along the X axis. So you are in effect not really defining the shape beyond its simplest basis. How does any of this define the length of the waveguide and the rate of expansion of the waveform?