<<Note corrections made in blue. Original text incorrectly referred to "inflection points" when it should have referred to "maxima/minima points".>>
...i agree with everything you said. when the forced frequency is the same as the natural frequency of the rifle barrel those wave peaks...
There's no "forced frequency". The "input" that causes the vibration is pretty close to an ideal impulse which contains all frequencies in theory. The resulting vibration will be due to the characteristics of the barrel (its impulse response) and there's not much you can do about that short of changing the characteristics of the barrel by making it longer/shorter or thicker/thinner, or restricting its motion with pressure points or tensioning devices, etc.
If you input a single frequency or set of frequencies rather than an impulse, you might, with sufficient amplitude, be able to force the barrel to vibrate at a different frequency than it "wants" to. However, that theoretical capability is pointless since a fired shot can't do that. Your "input" to the barrel is the discharge of the shot and that's not going to force anything. It's going to make the barrel "ring" at it's natural response frequency.
It's like ringing a bell. You hit it with a hammer/striker (an input impulse) and it rings at its natural frequency. You might be able to get it to resonate at some other frequency if you put another frequency in at sufficient strength, but what would be the point of knowing that if the only way you know to get it to ring is to hit it with a striker?
The idea behind the BOSS is that it allows the shooter to effectively change the barrel length (which alters the barrel's response frequency) so that for a given load the bullet exits at/near a
maxima or minima point in the muzzle vibration pattern.
...there is another thread currently running that is discussing why and how browning's BOSS system works. there has been a lot of conjecture and anecdotal evidence put forth (and a good reference to a 100 year old scientific study) on this, but no explanation seems reasonable or logical.
The BOSS is based on pretty basic system response theory. It may seem like it's not logical or reasonable, but that's only from the perspective of someone who doesn't understand (or has decided to reject) the principles which make it work.
the problem i have with your explaination is that bullets can't tell when the barrel is in the right position to print a certain way on the target. the barrel is going to move pretty much the same way every shot, but the bullets come out the barrel at different times, so a fast bullet may come out higher than a slow one.
That's why the BOSS doesn't work for every possible load--you have to tune it for each load. If you shoot a load that's a lot different, you won't get the same effect because the muzzle won't be at/near a
maxima/minima point when the bullet exits.
The reason for using the BOSS to tune the barrel so that the bullet exits at a
maxima/minima point is because that's the point where the barrel is moving the slowest during its vibration cycle. That means that the unavoidable small shot-to-shot variations in a given load will have a minimal effect on where the muzzle is when the bullet exits because the muzzle is moving relatively slowly during the general timeframe where the bullets will exit with that loading.
If you change to another load with a significantly different velocity without retuning the BOSS, the bullet almost certainly won't exit at a maxima/minima point and the normal shot-to-shot variations will now mean that the muzzle will be at a significantly different point for each bullet exit because the muzzle is going to be moving a lot faster during the general timeframe where the bullets will exit with the new loading.