See Figure 1: Selected Results with 1” or Less CTC Groupings

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If you’re wondering what the winner was, please know that I will get to that. The short answer is none of the above. The test patterns shown so far are anomalies and are the result of over-charging. If you want to see the sub-MOA results from my final selection, they’re at the end.

It took me from December 2015 to July 2016 to perform the range tests necessary to get the data needed to perform my analysis. This was a long, slow process. I was only able to work on it when I had time. I might not have done it at all without the additional encouragement and motivation from forum members. For that I need to acknowledge the suggestions and help I received from Smustian, Dellet, TMD, Plant.One, Rebel, and who could forget that loveable lug, Certifiable.

My goal was to achieve a more repeatable result than I had before. For that, the micrometer sizing die and a good Mitutoyo caliper became indispensable. There may not be such a thing as junk science, but there is such a thing as junk hypotheses and junk test methods. Without using a micrometer sizing die to resolve CBTO lengths to within a tolerance of .001”, I was grossly undersampling my data. I wasn’t able to make incremental changes in CTBO length. This had caused me to miss out on discovering a wildly exciting phenomena that was effecting my results. At least, it was wildly exciting to me because it was new. This phenomena is created by how the energy shockwave from the bullet’s discharge creates a standing wave, or rather a series of standing waves, in the barrel. At least a portion of the standing wave harmonics that effect load patterns operate in the ultrasound region. This region is defined as being roughly from 20 kHz to 200 MHz. My data indicates that there may actually be more than one standing wave and that competing standing waves operating over different effective lengths might actually be modulating each other. (See Figure 18). The higher order harmonics from these standing waves operate at frequencies that are beyond the range that can be captured by high-speed cameras, and consequently they cannot be resolved by methods for fully bedding a barrel. These high frequency harmonics create excitation at the near molecular crystalline lattice level in the metal alloy of the gun barrel. This was all new to me.

**The Pertinent Physics of Harmonic Motion in Waves and Waveforms:**

For standing waves and modulated standing waves in a gun barrel, the length of the gun barrel determines the fundamental wavelength. The length of the gun barrel is always half a wavelength. Any wave that moves down a finite length of barrel is always going to complete half a wavelength, or Pi radians, in its first fundamental period.

The highest frequencies I measured from my test data had half wavelengths on the order of .002”. In measuring waveforms, there is a limit to how high you can go. The rule states that you can only measure and detect frequencies that are no higher than those for which you can collect at least two equally spaced data points for each wavelength of that frequency. I now know that the ultrasonic phenomena that appears to be effecting pattern dispersion is undetectable unless you can resolve CBTO lengths to increments of .001”, or less, because the highest frequencies I was able to detect had half wavelengths of .002”. If the frequencies of this phenomena had gone any higher, meaning if they had any higher order harmonics, I wouldn’t have been able to detect them from CBTO measurements with a tolerance of only .001”.

With the micrometer sizing die, I was able to identify a series of harmonic wavelengths in the energy shockwave that effect pattern dispersion. These harmonics ranged from an estimated 676 kHz at the low end to 40.7 MHz at the high end.

**Test Conditions:**

Rifle: 300BLK semi-automatic CMMG Model No. 11098, 16.25” barrel. Fired suppressed using an SWR Octane 9 HD suppressor. The OEM adjustable buttstock was replaced with a more solid synthetic A2 style buttstock. Converted to a pistol gas system using a Wilson Adjustable Gas Block Kit.

Bullet, Powder and Primer: SMK 220gr, sorted into 0.1gr interval groups. IMR 4227, Wolf Small Rifle Magnum primer.

IMR 4227 may not be the best power to use for subsonic loads. I like it, but I’m not in a position to say how good it really is. More importantly, that’s what I had so that’s what I used.

What I lament is the general lack of an objective and quantifiable standard for burn rate charts. Knowing the order of burn rate for a series of different powders is nice but having a specific quantifiable rate of burn with the kinetic energy content would be better. Even better would be if burn rate information was coupled with quantifiable information on how consistent or how variable that burn rate is.

Charge Weight Measurement: I switched over to using a 1 milligram digital scale early on to take advantage of the increased resolution. Charge weights were measured out to +/1 mg, yielding a charge weight range per weight interval of only 3 mg. This was a significant improvement over my previous digital scale that only provided resolution to 1/10th of a grain.

Brass: Mostly once fired military .223/5.56 rounds converted and resized at home to 300BLK. High content of Lake City, Foreign NATO and Taiwanese.

Cartridge Trim Length: My stock of trimmed cartridges ended up including cartridges that were trimmed to two different lengths: 1.355” and 1.363”. In reality, trim length varied from 1.350” to 1.365” in a more or less normal distribution with most trim lengths centering around 1.355” or 1.363”. This is what I had so this is what I used.

Target Range: 100 yards with targets comprised of 1” square grids. Volleys were conducted with (5) rounds each. Two volleys per test point were taken on two separate days in preparing the graph of pattern dispersion as a function of CBTO length.

Operating Temperature: I learned early on that my barrel has two clearly recognizable steady state conditions. The first is cold barrel operation where I’ve fired less than (10) rounds through the barrel and the barrel is at or near ambient temperature. Firing (10) rounds or less through the barrel doesn’t increase the temperature of the barrel by more than 10 degF. I take my temperature measurements directly from the outside of the barrel at the top of the quad rail. Raising the barrel temperature by at least 10 to 15 degF causes my pattern to get wider or more dispersed, but this only lasts for a short period of time. This temperature induced wider pattern settles back down once the barrel temperature has been raised over 20 degF from it’s initial pre-firing temperature. That’s my rifle’s ‘hot barrel’ operating temperature range.. Once it has warmed up, it settles down and further increases in barrel temperature don’t effect pattern spread.

It was actually Dellet who turned me onto the fact that barrel temperature will not effect reloading tests. He was the one who told me not to worry about barrel temperature in conducting tests. What he was talking about was the fact that the velocity of an ultrasonic wave in an incompressible material like gun barrel metal doesn’t vary with temperature. The speed of sound varies in air with temperature but not in metal. The speed of sound only varies with temperature in compressible media like air. That means that hot barrel tests that are conducted after the initial thermal expansion to the barrel has been achieved, are just as valid and just as consistent as cold barrel tests, and will produce identical data, because the velocity of the energy shockwave doesn’t vary in metal.

See Figure 2: The longitudinal and transverse wave velocity equations on pg. 3 of J. D. Lavender’s “Ultrasonic Testing of Steel Castings,” Steel Castings Research & Trade Association, Sheffield England, Copyright by the Steel Founder’s Society of America, 1976. Excerpt reprinted here for academic purposes.

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From the wave velocity equations in Figure 2, it should be clear that sound velocity, i.e. energy shockwave velocity, are not functions of temperature in metal.

Of course, this creates an apple-to-orange comparison between my test results and my previous results. Previously, I used to cool the barrel down to its initial temperature at the start of each (5) round volley. I don’t need to do that now to achieve consistent results. That saved me time, but now I can’t compare my new results to my old 1.2 MOA best without first recognizing that it is an apple-to-orange comparison. The reason for this is that barrels shoot better when they’re cold. The reason has nothing to do with harmonic oscillation of gun barrel crystalline lattice structures in the ultrasonic region like what will mainly be discussed here. It’s the result of thermodynamics. Shooting patterns are more tighter and more consistent at lower temperatures. There is less chaos, or entropy, in the release of thermodynamic energy when that energy is released at a lower initial temperature. That’s why I take my hat off to marksmen like TMD who live in Texas where higher outdoor temperatures effect rifle performance. The reason is because higher temperatures create a greater threshold of thermal excitation. Charge release rides on top of that threshold. This is why extremely competitive marksmen refrigerate their ammunition before use, whenever practical. That’s something to know, not something that will be effecting my tests.

**Test Methods:**

I do not use a chronometer. I rely on my hearing to determine whether or not my shot has gone supersonic. My goal is to improve the tightness of my pattern. This does not necessarily require knowing the bullet’s velocity because I can tell what the tightness of my pattern is just by looking at it. For me, reducing pattern size is the goal, not getting a series of test shots with very little variation in velocity. Although velocity is important, I don’t necessarily have to worry about velocity to do CBTO tests. Velocity is not the end result I’m interested in. Pattern size is. If I can see what varying the CBTO length does to my pattern, that tells me what I need to know, or at least most of what I need to know. I will get into the rest of what is needed and explain this a little bit later. Before I do, I have to review the relative difference seen in pattern dispersions in the x (windage) and y (elevation) directions because I think I’ve figured out what’s causing them.

For now, let me point out what I already know from using the online Hornady Ballistics Calculator. First, from the online Hodgdon’s Reloading Data Center, I know that the book value for the muzzle velocity of my factory equivalent starting load is 1,044 ft/sec. That starting load has a relatively small chamber pressure of 26,700 PSI. Now I take that muzzle velocity and enter it into the Hornady Ballistics Calculator and vary it to see how much change in muzzle velocity is needed to get vertical spreads of ½”, 1”, 2” and 4” at 100 yds. To do this I use a zero range of 75 yds. to simulate being zero’d in at 100 yds. as much as possible:

See Figure 3: Online Hornady Ballistics Calculator Results for Determining Trajectory Drop as a Function of Muzzle Velocity

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I stopped after dropping the muzzle velocity only 100 ft./sec from my initial velocity of 1,044 ft./sec. The vertical drop in trajectory after lowering the velocity by 100 ft./sec was only 0.9”. That result is both obvious and telling. What this shows is that if I’m sighted in at 100 yds., instead of a mere 75 yds., in order to get 0.9” of an inch in vertical spacing, without some other forcing function present, I need a change in muzzle velocity of at least 100 ft./sec. If I’m seeing changes in my vertical spacing in excess of 1” for changes in CBTO length that are only about .001”, where charge weight is virtually constant, there is some other forcing function present. I’ll get back to this.

**Handling Fliers:**

From reading many posts I’ve learned that everyone is always concerned about how much significance they should attach to their fliers. The question always seems to be how much of a factor they should have in determining how good their pattern is. Some people think they don’t need to be counted. Others think they do. The general tendency seems to be to only include them if they don’t effect the result much.

It’s true that many know they need to be counted using some sort of weighting system but don’t know how. From a mathematical perspective they need to be counted, but with a weighting that’s less than full value. The reason is because their location and number show they occur less often than the larger number of bullets grouped together that they’re distanced from. Simply put, they are less likely to occur than the rounds that are clustered together at the center of the grouping. That’s the important thing to recognize because it’s the change in distribution that reflects the nature of the pattern’s spread that’s important statistically, not it’s actual size.

Because of this, I don’t use solid distances like CTC to make comparisons. CTC resolves into fixed x and y coordinates of GW and GH that don’t tell me anything about my pattern’s density or its distribution. That’s why I use a statistical comparison instead. The two best and easiest statistical comparisons are Average distance To Center (ATC); and Rsd, where Rsd is the probability of enclosure with a radius equal to the first standard distribution.

ATC is more formerly known as the probability of enclosure having a radius equal to the average distance from the center. This represents a radius of enclosure of 50%, i.e. the average. Similarly, Rsd creates a probabilistic circle of enclosure that’s 68%. This means that with Rsd, 68% of the rounds can be expected to fall inside the circle. The use of the Rsd number creates a better comparison between extremes of wider and narrower patterns than the ATC number does because it includes a higher percentage of hits inside its circle. That’s why I use Rsd instead of ATC.

This is one of the reasons why I don’t use ‘On Target.’ ‘On Target’ is great, but it doesn’t give me the measurements I want. Worse, having to drag the mouse all over the place makes it slower than simply typing the x and y coordinate values into an Excel spreadsheet while I’m looking directly at the grid, and Excel lets me make the calculations I want. That way I don’t have to take my eyes off the grid. I’ve been interpolating values between 0 and 1 visually my entire life. Distinguishing between 0, 1/4th, 1/2, 3/4ths and 1 is easy enough. My bullet diameter is roughly 1/3rd of an inch. That gives me a visual comparator for easily making measurements of 1/3rd and 2/3rds. On the bullet’s radius, the difference between paper fray mark and bullet smear is roughly 0.10” and 0.05”, respectively. That completes the visual clues needed to let me do what I do quicker than dragging a mouse all over the place.

Consequently, I can make bullet placement measurements quicker visually and I can do it to within a tolerance of +/- .05. That’s good enough for the rudimentary statistical analysis I use here. A higher level of accuracy than that doesn’t change the statistics enough to matter. Any differences are going to wash out anyway when averaged together so I don’t worry about them.

… So … That almost nearly explains how I handled fliers. For data collection in producing the primary CBTO graph, I included all fliers in calculating Rsd. Too many points on that graph had charge weights or CBTO lengths that were less than ideal and resulted in extreme values that couldn’t easily be attributed to anything else, so I included them. When I got to analyzing final results, I had to do things differently. By then, I could reasonably expect that my charge weights and CBTO lengths were going to give me results with a fairly good tightness of fit. By then, fliers were occurring less often and were more likely to reflect an inconsistency in the load, or a weakness in my shooting. Therefore, I established a personal rule for handling fliers in my final results. If the distance from the flier to the next closest member of the pattern was greater than the distance between the next two bullets in the grouping that were furthest apart by a factor of 2 or more, I discounted the flier as a fluke, attributable to a bad load or bad shooting. Only one pattern in a series of volleys representing a final result from the same day could have a flier discounted, and no pattern representing a final result could have more than one bullet treated as a flier and discounted.

**What Was My Goal:**

My goal was not to reduce the size of my pattern. That would be nice, and I hoped it would happen, but my goal was to reduce the distribution of my pattern instead. That could reduce the size of my pattern, but not necessarily. The reason is because my goal was a probabilistic one, not a deterministic one. For this reason, I did not use CTC as a metric in comparing the relative pattern spread for different loads. I only used it at the end of this writeup to illustrate my results. Let me provide an illustration so that you can see why I did things differently.

Take a look at the following test patterns:

See Figure 4: Sample of Simulated Test Patterns all having GH = GW = 1, and Where All Rounds Are Uniformly Spaced.

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Please notice how these samples represent the full range of uniformly spaced patterns that can be achieved where GH = GW = 1”. I recognize that with GH = GH = 1 that the CTC will end up being 1.4, which is not sub-MOA, but that’s besides the point. You can always scale these patterns down to get CTC = 1” if you want, but the point remains. The point is that when your goal is to achieve a certain set of GH and GW values so that all your shots fall within a fixed aperture, you can end up with all sorts of patterns whose density, distribution, or tightness of fit can range anywhere from being great to abysmal. This is because even though all these patterns have a GH—GW value of 1”, their Rsd values can range anywhere between about .71” at the high end to about .50” at the low end. Tighter patterns have lower Rsd values even though their GH and GW values are still 1.” In other words, GH, GW and CTC tell you nothing about the pattern’s tightness or density. All they do is give you a fixed aperture for fit.

My ‘pre-micrometer sizing die’ best was an Rsd of .57”. I never bothered to check to collect data to see how repeatable that load was. I knew it wasn’t very repeatable and I left it at that. Therefore, my new goal became to see if I could achieve an Rsd less than .57” that was repeatable. When I get to my final results, I will show them using CTC since that’s the figure of merit everyone seems to use. In the meantime, all of my analysis will be done using the statistical value of Rsd. If there’s a pattern in the data, it will show up better that way. Wave generated harmonic functions, thermodynamic functions, and all other functions including those with multiple modalities, are more probabilistic than they are deterministic and are more easily modeled that way.

See Figure 5: ‘Pre-Micrometer Sizing Die’ Best

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**More on Test Methods**

For anything that is being perturbed by a set of harmonics, the best results are always going to be at the end points of the wave equation where all harmonics fade to zero. Given this, the natural place to start testing for best CBTO length would seem to be at a CBTO length that’s approximately equivalent to maximum COL.

I chose not to do this for several reasons. The first reason is because I wanted to collect at least one wavelength’s worth of data for the low order harmonics that were present to see what sort of pattern existed. On a waveform like barrel harmonics where only the magnitude of a phenomena like pattern dispersion can be measured, one wavelength is two half wavelengths or two relative maximums and two relative minimums. I needed an interval of CBTO lengths like that, culminating at maximum COL, to give me the data I wanted.

I wanted to get experience taking measurements with the bullet comparator on my dial caliper. I wanted this practice to make sure I was making consistent measurements before I got to that part of the test interval near maximum COL. I didn’t want early measurements to compromise the results at longer CBTO lengths because those were the ones that I expected to yield the best results. That was a natural bias that I went in with. Having preformed opinions of what to expect is not all that scientific but it is human nature. It’s hard to remove so I simply went with it and recognized it going in by starting at the opposite end.

Finally, I had no idea what sort of charge weight equivalency curve I should use for any CBTO length under test. That motivated me to start at the opposite end from where I expected to obtain the best results. That way if my first attempts at producing a decent charge weight equivalency curve were flawed, it would have the least effect on the charge weights I might ultimately use for my final tests. Another more direct way of saying this is that I wanted to see what the effects of over-charging would have on my initial results. These were results I didn’t expect to care about. Consequently, I started my testing with a CBTO length of 1.512” and a charge weight of 10.1 gr., and worked up. Initially, I kept charge weight constant.

As you recall, each Rsd value was based on a (5) round volley. Ultimately, I ended up taking the average of two volleys for the CBTO intervals of the greatest interest and using that as the point for that CBTO length. For most of my early testing, I did not vary charge weight over given intervals. Instead what I did was to use Excel-generated trendlines for intervals of the CBTO dispersion curve to determine what my relative level of charging was.

It wasn’t a foolproof method, but what I discovered was that over-charging didn’t change the average Rsd for a given section of data. What it did instead was to increase the modality of the Rsd dispersion curve. To understand why required that I do further research into the propagation of energy waves in solid media like metal. In incompressible media like gun metal, there are two separate energy shockwaves. These two shockwaves are known to be loosely coupled depending on the material’s cross section. Consequently, they typically have different velocities. There’s a longitudinal wave and a transverse wave. Because gun barrels are round in cross-section, it’s reasonable to expect the coupling between the longitudinal wave and the transverse wave to be consistent. Because they are operating in the ultrasonic region, the energy waves or shockwaves themselves don’t necessarily create vibrations that result in harmonic motion. The energy waves with lower frequencies in the sonic range do that. Instead, these ultrasonic energy waves are creating excitation in the near molecular crystalline lattice structure of the metal in the gun barrel.

Doing the math on the exponential decay rate showed that the longitudinal shockwave that runs the length of the barrel die out to nothing by the time the bullet leaves the barrel. In any case, the longitudinal wave doesn’t create excitation that could transfer energy into the x, y plane perpendicular to the barrel. In contrast, the transverse wave operates in a plane that is perpendicular to the barrel. It perturbs things in that x, y plane. It is the transverse shockwave that is responsible for pattern dispersion.

All energy waves have modes. Usually, these modes cause variations in the geometric cross-section of the media they are transferred through. This doesn’t happen in solid, incompressible media like gun metal. Instead the various modes trigger various excitation levels. These excitation levels translate into amplitudes, or measurements of dispersion. Even though they travel at the same velocity, shockwaves with greater energy have more modes of propagation than shockwaves with less energy and show more modalities with higher amplitudes. These additional modes represent greater volatility, or higher highs and lower lows in the dispersion pattern. Consequently, it is possible to get an occurrence of an extremely low Rsd from a poor CBTO length due to over-charging. These kinds of results are not very repeatable. Part of the goal was to identify CBTO lengths and charge weights where favorable results would be more consistent, and thus not likely to be the result of over-charging. Therefore, it was important to be able to recognize what over-charging looked like.

In contrast, under-charging occurs when the charge weight is less than ideal for that CBTO length. When the charge weight is less than ideal, there is insufficient charge to drive the bullet with an optimal full force through the barrel. Under all conditions, not all of the potential energy in the charge gets translated into kinetic energy for driving the bullet through the barrel. Some of the charge’s potential energy is absorbed by the impact of the bullet being seated on the rifling. When the charge is less than ideal, more of a percentage of the charge ends up being absorbed in seating the bullet onto the rifling. This higher percentage of absorbed charge leads to higher levels of inconsistency in the amount of energy transferred to the bullet as kinetic energy. Consequently, under-charging also produces modalities that create wider patterns and greater dispersions. The only difference between under-charging and over-charging seems to be that the extremes created by under-charging are not as great as those created by over-charging.

So, my primary test method would be to plot the dispersion or distribution of my pattern, as measured in units of Rsd (inches), as a function of CBTO length. The interval would be from 1.512” CBTO to maximum COL. Test intervals would be in increments of .001”. Figure 6 shows the equivalency formula I used for maximum COL.

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That became the end point for my tests. Happily, I discovered that with the consistency achieved with the micrometer sizing die, I was able to cycle rounds at that length from my magazine.

Even though the plot of dispersion as a function of CBTO length is not a true wave, it still represents a waveform that can be subjected to waveform analysis. A wave is something that varies with time. A waveform is a piecewise linear construct that can be analyzed using the same mathematical tools as a wave. This plot meets the criteria for being treated as a waveform in part because it’s measured in incremental units of distance, and:

Incremental change in distance = incremental change in velocity * incremental change in time, i.e:

ds = dv * dt

On an incremental basis, this is a repeatable, linear relationship. By repeatable, I mean that the data is repeatable on a statistical basis with a reasonable level of certainty. This is another reason why the data to be used had to be expressed as statistical values.

Even though the units represented by time and velocity are constantly changing, i.e. accelerating, over the length of the bullet’s travel through the barrel, they are always inversely proportional to each other for incremental units of distance. Since the units of distance I will be using are uniform and evenly spaced, there’s no problem with linearity in scaling. Nothing gets lost by that. This makes this function eligible for being treated as a waveform because it is created from a series of linear, incremental distances where the dispersion function in question varies predictably with those incremental distances. Consequently, all the mathematical tools created by Gauss, Euler, Laplace and Fourier for time domain and frequency domain analysis to processes involving harmonic motion apply and can be used.

The big question will be whether or not this waveform will help to explain the behavior created by small changes in CBTO length, and whether or not a mathematical model can be created of that waveform to predict its behavior, or at least explain it.

**Early Results:**

See Figure 7: Xsd and Ysd as a Function of CBTO Length

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What this graph shows is the cyclical change in dispersion pattern of Rsd after it has been split into two component vectors, the x and y components that are at right angles to each other. The x component represents the vector along the horizontal axis. The y component represents the vector along the vertical axis.

I promise you I did not train my shooting arm to waiver or move more or less erratically in a cyclical fashion as I varied CBTO length. This is a strong indication that the conventional wisdom that the supposed deviation in windage due to trigger pull is not well grounded in objective science. Instead, it suggests that the transverse wave is rotating as it follows the longitudinal wave down the barrel in a cyclical or sinusoidal fashion.

This graph shows a number of other things as well. Before getting into the rest of that, let me first point out that we can only measure the magnitude of the transverse wave, not its angle. That’s why the sinusoid for barrel harmonics repeats itself every half wavelength instead of every wavelength. However, this graph is the first indication I’ve had despite all the literature I’ve reviewed that the transverse wave in barrel harmonics is a rotating vector. This makes sense because there is no minor axis in a barrel with a radial cross-section. That means all modalities have no choice but to manifest themselves equally as radials across the cross-section of the barrel. I believe this would be true for octagonal barrels as well, but have nothing to support it.

Therefore, all modalities have to exhibit themselves in the cross-section evenly and radially. As we’ve seen, the two component vectors Xsd and Ysd from the Rsd waveform are generally 90 degrees out of phase with each other, but not always. There are some subtle changes in the phase relationship of these two waves. They don’t stay 90 degrees out of phase all the time. They move in and out of phase and are mostly 90 degrees out of phase. It took me a long time to realize this but the reason is because the Rsd waveform is comprised of harmonics whose amplitudes change and vary at different rates. This causes a perception of shifts in phase between Xsd and Ysd, but this just an illusion. It’s occurring because I can only plot information on the relative magnitude of Rsd. I’ve never been able to collect data to show its relative phase.

**Why Is Ysd so Much Greater Than Xsd**

The next question is why is the Ysd component so much greater than the Xsd component. The answer to this question came to me quicker than the previous question. If you think about it for a minute, anyone who has ever been on a rollercoaster will know the answer. Remember back when you were on an old time track roller coaster whose rails weren’t lined up very well. When you go across track segments on a flat curve that are not aligned in the side-to-side or x direction, you get jostled around a little bit. In comparison, when you go across rollercoaster track segments at the bottom of a dip or the top of a hump that are not aligned well in the up-down or y direction, you really feel it. In fact in comparison to the little jostles you get in the x direction going around curves, those jostles in the y direction coming up or doing down can be painful. The reason is because gravitational forces are either accelerating or decelerating the instantaneous forces created by the track segment up-down discontinuities. This is most noticeable when they’re occurring in synche with the position of the rollercoaster at the bottom of a dip or the top of a hump.

I believe the same thing is happening to the bullet in any gun barrel. If the radial direction of the transverse wave is aligned vertically when the bullet leaves the barrel, the transverse wave is going to either amplify or mitigate the force of gravity. If the bullet is riding the top half of the barrel at the point of departure, g-forces will be amplified by the transverse wave and cause vertical stringing. If it departs off the bottom half, the g-forces will be mitigated but still be greater than when the transverse wave was horizontal.

The phenomena of vertical stringing can’t be explained properly by the normal distribution of muzzle velocities for a given load unless the trajectory drop due to the variation in those muzzle velocities is what was shown earlier with the Hornady calculator. For my load, a muzzle velocity variation slightly greater than 100 ft./sec is required for every 0.9” of vertical pattern spread when performed at a sight in distance of 100 yds. Based on experience, that doesn’t seem likely, and I’m forced to conclude that I’ll get vertical stringing every time the transverse wave is aligned vertically at the point of bullet departure. This will be especially true when the bullet is riding the top half of the barrel and is being slammed back downwards by a transverse wave that’s being accelerated by gravity.

**Test Results: The Plot of One Wavelength’s Worth of Test Data**

The following chart required the expenditure of more than 1,500 rounds to prepare and countless hours at the range. My thanks goes to the Gig Harbor Sportsman’s Club and its Range Officers for providing an exceptional shooting environment in which to conduct these tests:

See Figure 8: Pattern Dispersion, in the Form of Probability of Enclosure Radius (Rsd), as a Function of CBTO Length

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Once I had prepared this curve, I realized I had seen something like it before. It’s a representation of a sum of harmonics in a series. As a comparison, this is what the first (3) odd harmonics in a series look like:

See Figure 9: First (3) Odd Harmonics in a Series

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If those harmonics are evenly weighted and added together, they look like this:

See Figure 10: Summation of Evenly Weighted Harmonics, N = 1, 3 and 5

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You will notice that all of these charts cover a length from 0 to Pi, which is half a wavelength. Every barrel, or reflection length, is always half a wavelength for the harmonics it generates. This means that any mathematical model represented in units from 0 to Pi explains the behavior of one gun barrel just as well as it explains the behavior of any other gun barrel, assuming the nature of any reflection points, if any, match in terms of their length and position. Thus, for purposes of making a mathematical model the only thing that needs to be done is to have the barrel’s effective length converted into units of radians.

After reviewing some of the curves for the summation of weighted harmonics, I decided to see if I couldn’t come up with a mathematical model that represented the data I had obtained. First, I selected a specific part of the curve that best represented my data. This happens to be that portion of the curve from 15/16 Pi to Pi. I used curve fitting techniques and adjusted exponential decay rates to recreate the perceived effective decay rate.

I adjusted the harmonics used for the summation of the waveform as best I could. My goal was to identify a sum of a series of harmonics which would recreate the shape of the curve in my data as well as possible. What I discovered was that the first order harmonic essentially only establishes the slope of the trendline for the interval in question. Refinements changed the interval in question to .9455 Pi to .9925 Pi. Since I had already found a good fit for the effective decay rate without the first order harmonic, I left the first order out so as to avoid having to recalculate a new decay rate for it. A purist would have put it back in and started over. What I found out, given that adjustment, was that the series of odd harmonics from (3) to (13) was the best fit. Odd harmonics represent the contribution to the waveform from energy reflecting off the end opposite the starting point. Even harmonics, starting with the fundamental or N = 0, represent harmonics from the energy reflecting off the backstop behind the starting point.

Finally I translated the barrel length in radians to effective CBTO units in increments of .001”. This graph is only predictive up to maximum COL.

See Figure 11: Partial Sum of Harmonics for Simulating the Test Data on the 300BLK Load

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Here is a side by side comparison of the experimental range data to the mathematical model comprised of harmonic components. These are the harmonics that are believed to be contributing the most to the effects of the transverse wave.

See Figure 12: Side by Side Comparison of the Experimental Data to the Mathematical Model for Representing the Impact of the Harmonics from the Ultrasonic Shockwave

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**Differences Between the Math Model and the Experimental Data:**

There are some differences between the experimental data and the math model, but I think they can be explained. There was clearly some over-charging in the early part of testing. That would explain the more extreme variations in the experimental data around what are roughly the same average Rsd trendline values in the mathematical model. Over-charging results in a greater number of modes, or a higher level of modality. This leads to more extreme highs and lows in Rsd values. Neglecting the contribution of the high frequency harmonic components to the Rsd graph, I simulated what those extra modes do to the trendline in the following graph:

See Figure 13: Projected Modality Envelopes for Charge Weight Equivalency Comparisons

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**Charge Weight Equivalencies:**

Here is the chart that shows the various charge weights I used at various stages of my testing:

Figure 14: Chart #2: Estimated Charge Weight Equivalencies as a Function of CBTO Length

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My first testing used the charge weight equivalency curve shown as a dashed green line labeled ‘Prelim 1’. One set of tests went from 1.512” to 1.561” using a charge weight of 655 mg. The next half of this first set of tests went from 1.616” to 1.664” using a charge weight of 679 mg. The trendlines from these two sets of tests both sloped downwards towards each other and intersected at a point at 1.594”. The average of those two charge weights was 667 mg and so, flawed as it was, that became my first point for charge weight equivalency.

My second set of tests used the charge weight curve shown as a green dot-dashed line labeled as ‘Prelim 2’.

My third set of tests was done using a sampling of points not previously tested using the charge weight equivalency curve shown as a red dotted line labeled as ‘Prelim 3’.

By looking at relative trajectory drop, both the Prelim 2 and Prelim 3 tests showed I was under-charging at points near maximum COL. Since I already had my precious curve, from there on out I simply walked the charge weights up for two specific CBTO test points, 1.657” and 1.663”.

There is something to be said for using a constant charge weight to produce a linear trendline for determining charge weight equivalencies as CBTO length is varied. Theoretically, it would give you exact results. Of course, for that to happen, the interval spanned on either side of the intersection point would have to be the same length; and, if there is an inflection point or change in the curve, that relative minima or maxima would have to be the same distance from the closest end of each of the spanned intervals for this to work exactly. Since I didn’t know where my inflection points were when I started, I pretended not to care. What I knew was that the results of over-charging or under-charging would show themselves as modality extremes and that I could go from there.

The net result is shown in Charge Chart #4:

See Figure 15: Chart #4: Finalized Charge Weight Equivalencies as a Function of CBTO Length

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Once again, I have to hand it to Dellet. It takes more than intelligence to come up with a paired set of (CBTO Length, Charge Weight) values that line up exactly to the onset of supersonic speeds at maximum COL for a random load, but he did. That came from the thought problem he posed to me. Lining numbers up like that takes more than intelligence. It takes experience and skill. Unfortunately, when I got to that point in my testing where I was testing best CBTO lengths near maximum COL, I wasn’t looking for the threshold of supersonic. I was looking for something that doesn’t show up on a chronometer. I was looking for the beginning of the transonic region. It’s effect on pattern dispersion is clear. It is one of the features of aerodynamics near supersonic flight that has to be taken into consideration.

The following graphs show what I found when I tested various charge weights at my last two CBTO length contenders. The first curve is for a CBTO length of 1.657” The second curve is for a CBTO length of 1.663”:

See Figure 16: Dispersion Pattern in Rsd as a Function of Charge Weight at CBTO = 1.657”

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See Figure 17: Dispersion Pattern in Rsd as a Function of Charge Weight at CBTO = 1.663”

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It was at this same point in my early preparations in going over the nature of charge weight equivalencies where Smustian emphasized something very important. Establishing a charge weight equivalency curve needs to be done to come up with final load values. It’s an important check on the work that’s been done. However, by identifying the pattern of behavior over the full length of my test interval first, I was able to complete my charge weight equivalency curve merely by finding the best charge weights for the two CBTO lengths that had already demonstrated the best results. The location of these CBTO lengths is, of course, the closest relative minimum created by the highest frequency components of the shockwave that’s nearest to maximum COL. Creating a picture of the general trend of my dispersion curve insured being able to see that visually despite the modality extremes in the over-charged areas at the beginning and the under-charged areas at the end of my dispersion curve testing.

In other words, by using more bullets I ended up finding the same result. Someone had to come up with a clear written explanation and post the result, so it might as well be me. This twenty question questing game has gone on long enough. In today’s world with today’s politics, responsible firearm owners need rapid access to the information necessary to produce an accurate load. It’s a matter of civic duty.

**The Threshold of Supersonic**

One other interesting thing can be observed from the graph in Figure 17. The nature of supersonic ballistics operating at the threshold of the supersonic region is clearly shown here. An initial supersonic velocity shows up at the beginning of a linear interval of increasing charge. This occurs at or near the beginning of the transonic region. In this case, the first unmistakable and tell tale sound of supersonic velocity showed up at 734 mg on the charge weight curve for CBTO = 1.663”. From 734 mg on, small increases in energy didn’t increase velocity, or even maintain it - at least not right away. Instead, a small increase in energy had a smoothing effect that helped to lower Rsd, while velocity went back down below supersonic. A continued increase in energy didn’t increase velocity until there was sufficient energy to support supersonic velocity at a new energy threshold that was higher than the original energy threshold. The difference between these two thresholds is what defines the transonic region. This region is a very complex region. It doesn’t show up on any chronometer as any set of specific readings. Instead, it’s defined by the shape of the projectile and the density of the atmosphere.

So anyway, after the higher energy threshold required for sustained supersonic velocity is achieved, both velocity and the pattern’s dispersion start going up again, this time together in unison. That marks the end of the transonic region and the beginning of completely unperturbed supersonic flight.

The transonic region is not the best region for minimizing the effects of turbulence on either aerodynamic or ballistic flight. In my opinion, this effects the region on the charge weight equivalency curve on either side of the initial relative maximum Rsd point where the energy level was sufficient for initiating supersonic velocity. Not surprisingly, it turns out that the best relative minimum for Rsd on the charge weight curve turned out to be located just below this region.

**A Closer Look at the High Frequency Data That First Had Me Puzzled**

The one question that had me puzzled at the beginning was where the high frequency harmonics in the pattern dispersion were coming from, as shown in Figure 8, the graph of Rsd as a function of CBTO length.

Like many people I’ve heard the repeated mantra of how the gas port in a semi-automatic creates wider patterns than what can be achieved with a firmly bedded, heavy barreled bolt action rifle. If you hear something long enough, you begin to believe it and you stop questioning what sort of objective data might actually be needed to prop that particular belief up. At times, I’ve been guilty of subscribing to conjecture just like that.

It is because of this legacy that I naturally assumed the high frequency harmonics I was seeing, as reflected in pattern dispersion, were being caused by the gas port and it’s related blow-back plumbing. This, after all, is the only thing that’s different between a semi-automatic and a bolt action rifle. Of course, I was wrong. It took Dellet’s unabashed comments and critical thinking to get me to question this.

Once I realized that the summation of natural harmonics in the transverse wave was responsible for my pattern dispersion, I tried to produce a model to explain this phenomena. That lead me to measuring all the half wavelengths that could be discerned in my test data to see what they could tell me. I was able to identify (6) half wavelengths of .002”, .003”, .004”, .005”, .007” and .011”, respectively, as higher order wavelengths. Half wavelengths of .0975”, .1110” and .1205” showed up as lower order wavelengths. The lower order wavelengths most closely resemble the general curved trendline, but the higher order wavelengths provide information on both harmonic generation and multipath modulation.

Since there are more repetitive instances of the higher order wavelengths, they are the wavelengths that are going to show the most pronounced effects of modulation. Modulation will have more of a visible influence on them. Certainly, one can observe that the sum of some of these values represents the sum or the difference of some of the other values. That’s the effect of modulation. That indicates modulation is occurring. This means that there are multiple standing waves at work that are resonating over different path lengths. Certainly different path lengths exist because there are different partial reflection points along the receiver, the barrel, the gas port and the suppressor. That lead me to try to see if I could separate the wavelengths out further by harmonic order for each possible firearm component or interval to see what the harmonics for each of those standing waves might look like.

The results for calculating out all the N order harmonic half wavelengths for various firearm components and intervals are shown in the following graph.

See Figure 18: Ascending N Order Harmonics Half Wavelengths for Various Firearm Components

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These graphs show that both harmonic generation and modulation are possible for various components because representative wavelengths that match these various components show up in the test data. If there are wavelengths that can be recognized as coming from different reflective paths, modulation between those standing waves has to be occurring. However, it becomes difficult if not impossible to tell which of these higher order wavelengths is a result of N order harmonic generation and which is a result of modulation from a strict time domain analysis. A different type of analysis is needed but I lack the tools for performing an FFT on the data.

However, some things are already clear. Since the lowest half wavelength I recorded was .002”, I was able to use that figure to figure out what the highest order harmonic was in the sum of harmonics that are perturbating my patterns. I know the shockwave responsible for this is going to be modeled for the most part by a sum of harmonics because of the law of conservation of energy. All the ultrasonic energy that isn’t being absorbed by the receiver or filtered out by the barrel is going to have an effect on the bullet regardless of what harmonic order it belongs to. This means that the harmonic orders that are closest to the bullet when it leaves the barrel are all going to be added together as a result of the law of conservation of energy. Thus, the mathematical model is represented as a sum of frequency components.

What’s more, this energy only has an effect on the bullet while it’s still in the barrel. Since I know, at least approximately, how long the bullet is in the barrel I can come up with an approximate velocity for the transverse wave.

Notice that the 13th harmonic order has representative values near .002” for the barrel, the interval from the gas port to the suppressor, the suppressor and the barrel, the barrel and the receiver, and finally, the interval spanned from the receiver to the end of the suppressor. Given the velocity of the transverse wave, that’s not too surprising. The values shown for the 13th order harmonic for these various span lengths are too close together to differentiate. This means there is a possibility that partial reflections of shockwave energy are occurring throughout all of these intervals, at each of these reflection points.

However, now I know how fast the transverse wave is moving. It’s moving (13) times faster than the average speed of the bullet traveling from the chamber to the end of the suppressor. Instead of worrying about trying to calculate what that average bullet speed is, I simply chose the initial factory load’s published muzzle velocity of 1,044 ft./sec as a reasonable approximation. That’s going to be on the high side of an average velocity, but it’s close enough for government work, especially since all I am trying to do is to produce a model for explaining things as opposed to outright predicting them. It follows that the transverse wave velocity in the metal alloy of my gun barrel is approximately 13,572 ft./sec or 4,136.75 m/sec. This falls within the range of book values I found for transverse wave velocities for various steel alloys.

Referring back to Lavender’s work on ultrasonic testing on pages 3-4 I found more useful information for determining whether the gas port is causing problems not faced by bolt action rifles.

See Figure 19: Lavender’s Work on Ultrasonic Testing, Page 4.

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After providing the reflection coefficients and the equation that proves it, Lavender clearly states that the reflection at an air to metal boundary is 100%. The hole in the gas port is the only place along the barrel’s effective length where there is a reflection due to an air-metal boundary. From there, it was easy to obtain a reasonable value for the reflection coefficient at the gas port. The hole for my gas port is 0.125” in diameter. The circumference of my barrel at the gas port location is 2.46”. This yields a wave reflection coefficient of 5%, which translates into a reflected power coefficient of a mere 2%.

A reflected power coefficient of 2% is practically nothing. The fundamental in a summation of harmonics based on 2% reflected power is insignificant by itself. The decay rate’s effect on the higher order components of such a small reflected power coefficient makes it even less so. That solved it for me. This means that the gas port is not creating a significant reflection point for energy shockwaves.

At the same time, I still cannot distinguish between frequency components of the shockwave to determine with any degree of certainty which ones are natural harmonics and which ones are the results of modulation. I can look for sums and differences of frequencies but frequently these values match the same values being generated by harmonics from other standing waves.

However, I can say that since the mathematical model resembles the test data fairly closely, that the effects of modulation from multiple standing waves only makes a small contribution to the dispersion curve.

In other words, one way to see whether or not other standing waves exist from other partially reflective points, is to look for differences between my mathematical model and the experimental data, to see if there are any. However, that runs counter to the mathematical model’s original purpose. The mathematical model’s original purpose was to model dispersion by manipulating the natural order harmonics to make it resemble the curve from the test data as nearly as possible. It was created by modifying the harmonic content of a single standing wave to fit the trendline of the curve the test data produced. That modification was created by finding an effective decay rate and by selecting harmonic content that matched. Harmonic content that could be attributed to the reflective energy from the muzzle end seemed to provide the best fit. However, in creating a best fit, I could also have been unknowingly modeling the effects that can still be attributed to other metal-to-metal partial reflection points. These are the shortcomings that are encountered when analyzing discrete waveforms in just the time domain.

However, there’s already enough data to show that multiple partial metal-to-metal reflection points along the effective length of the barrel exist. Even though they exist, their contribution is extremely small because the mathematical model represents the test data quite well. Finally, evidence has been provided that shows the effect of the air hole at the gas port is insignificant.

**Test Results from the Final Load Selection:**

See Figure 20: Test Results for the Selected Load, i.e. CBTO = 1.663” with 716 mg Charge Weight

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These target images show the following:

14 May 2016 Test Volleys

1. Rsd = .49” CTC = .997” or .95 MOA

2. Rsd = .51” CTC = 1.075” or 1.03 MOA

23 July 2016 Test Volleys

1. Rsd = .43” CTC = .93” or .90 MOA

2. Rsd = .70” CTC = 1.54” or 1.47 MOA (erratic – known shooter error)

3. Rsd = .45” CTC = .97” or .93 MOA

Rsd(avg) = .51” CTC(avg) = 1.10” or 1.06 MOA (With Worst Group)

Rsd(avg) = .46” CTC(avg) = .993” or .95 MOA (Without Worst Group)

Please remember that these are the results of ‘hot barrel’ tests. On average, I have always been able to achieve a nominal reduction of .15” in Rsd when I cool my barrel back down to its original temperature at the beginning of each volley. This is a thermodynamic issue not a reloading one. In any case, this nominal conversion factor of .15” Rsd allows for a better comparison between my original best of .57” Rsd, than a apple-to-orange comparison would otherwise allow.

The current average Rsd value of .46” has a standard deviation of +/- .03” Rsd, so the goal for better repeatability has been achieved, and sub-MOA accuracy can be claimed.

**For the Frequency Domain Analysis, See**:

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**:**

*Lest We Forget*Forumhttps://www.facebook.com/groups/743252869163379/