The following link's information is technically correct:
http://www.snipercountry.com/articles/realbenefitsbarrelfluting.asp
It’s not theory at all. Those 4th order equations are used all the time by engineers calculating rigidity for all sorts of round shapes of different materials.
But it's hard for most folks to understand. So, I'll try (probably the humpteenth time) to explain it it simplest terms.
* First, a popular concept....
Fluting a barrel makes it stiffer.
If you really believe that, you must also believe a patio deck made with 2x6's fastened together standing on their narrow 2" thick side can be made stiffer by replacing every other one with a 2x4. This equates to the flutes at the top and bottom of a barrel.
And you'll also believe a 6x6 beam made with three 2x6's laid flat atop each other can be stiffened by replacing the middle one with a 2x4. This equates to the flutes on the sides of a barrel.
Therefore, you must also believe if you remove material from something, it will bend less. Right?
* Second, grade school physics....
A long round tube in the horizontal position has half its mass above its center and half below it. Fixed at one end and a weight hung on the other end makes it bend down at its free end. The material in its top half gets expanded by some amount depending on its ability to stretch. And the material in its bottom half gets shrunk by some amount by its ability to be compressed. Same thing happens if it's bent sideways but 90 degrees away; one side compresses and the other side expands. If you remove any material, there'll be less to resist expanding and compressing.
* Third, changing those 2" thick boards to smaller metal pie slices....
Imagine a 0.9 inch pie made of steel that’s 24 inches thick. First, cut that pie into 12 slices. Then cut .15 inch off the slice’s inner tips to make a 3/10ths inch hole in the middle of that 24 inch thick pie. Bond them back together. Now you’ve got something like a 30 caliber barrel 9/10ths inch in diameter and 24 inches long made with twelve pie-slice shaped “wedges” of steel that are 3/10ths inch thick (bore diameter to outside barrel outer diameter), a bit less than 1/4 inch thick at their widest point (barrel outside circumference) and a bit less than 1/6th inch wide at their thin tip (bore circumference). ‘Tis nothing more than taking those boards in the above example and wrapping them around some axis.
Remove .15 inch from every other slice’s thick end and these six slices will now be a bit less wide at their thickest part and not as long from their inside edge to their outside surface. And this equates to replacing every other one of those 2x6's with 2x4's.
If you take the material cut off of those sliced and put it atop the uncut ones, those longer slices will have more area away from the pie’s center. That’ll increase the material that resists compression or expansion making those slices stiffer. Then the pie (barrel?) will be back at its original weight but will resist bending with more material further away from center. Same thing would happen with the deck if 2x4's alternated with 2x8's. Deck weight would be the same as if all 2x6's were used but a lot stiffer with more material added atop the widest boards. A board 1 inch wide and 1 inch high isn’t nearly as stiff in the vertical axis as a 1 inch wide board that’s 2 inches high
* What happens when you flute a finished barrel?
The stress on the metal makes it move from the flute’s cutting blades going into the steel. The internal diameters under the flutes will change a tiny bit; enough to be measured with precision tools such as air gauges and deep hole micrometers.
Button rifled barrels; the diameters get a bit larger, hammer forged barrels a bit smaller and cut rifled barrels - little, if any change. Minimal stress is the result as the metal’s scraped off in microscopic thicknesses in cut rifled barrels.
Unless you shoot your stuff into no worse than 1/3 MOA at 100 yards, you may not notice any accuracy degradation from fluting a finished barrel. But if the most uniformity in bore and groove diameters is wanted, flute the barrel blank before it’s rifled. Then lap the bore to uniform dimensions.
* In conclusion......
Few match rifles have fluted barrels because they do nothing to improve accuracy nor prevent changes in point of impact as they heat up. Good and even average quality barrels will not change point of impact as they heat up. Even when fired several times a minute, if properly fit to the receiver, impact doesn’t change. But as most folks reporting point of impact change with barrel heating up are using commercial factory rifles whose receivers’ face isn’t squared up with the barrel tenon axis, the high point on the face puts as stress point on the barrel and shot impact strings in the axis that point is from bore center. Cheap barrels that aren’t properly stress relieved will walk shots when they heat up regardless of how they’re fit to the receiver.
As a fluted barrel is less rigid than before fluting, its resonant frequency will be a little bit lower. Its harmonic frequencies are even multiples of its resonant one; second harmonic twice the resonant frequency, third is three times, fourth is four times and so on. Most barrels’ resonant frequency is less than 100 Hz, most in the 50 to 70 Hz range. Whatever its frequencies are, they are exactly the same for every shot fired regardless of the load used. Heavier loads make the barrel wiggle and whip greater amounts than light ones, but the frequencies stay the same. And for each successive harmonic frequency a barrel has, the amount of wiggle and whip for a given load gets less and less.