Thank you Lefty,
  So you are sure it is a certain fact that the barrel flexes up and down because of gravity?

The barrel vibrates horizontally in addition to vertically, but the amount is much smaller than the vertical because your shoulder (the support) is below the bore line. This produces a recoil torque (recoil force x distance from the center of the butt to the bore line)which causes the barrel to rise. The upward motion feeds energy to the vertical vibrations but not to the horizontal or the rotational.

Cat_Whisperer wrote on Mar 31^st, 2014 at 1:01pm:

Just so you know … no one seemed to believe ‘me’ – haha! - when I stated that barrels vibrate in the vertical plane due to gravity as the largest influence (less the shot fired). So I’ll quote Bill Calfee, who has written more than a few articles for Precision Shooting about barrels, barrel tuners, and barrel vibrations. In fact, he takes credit for introducing the world to barrel tuners via a 1994 article written about them, as recorded in PS magazine.

Thus I’ll quote him directly … emphasis mine :

”A rifle barrel is usually discharged in a horizontal position ... this means that the forces of gravity influence the vibrations of the barrel. This means the bulk of the vibrations of a rifle barrel, when fired, are in a vertical plane.”

This is how a barrel vibrates:

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And NOT like this … 

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Get your bullets to depart the muzzle during the node ... and you'll have more inherent accuracy, specially for any long-distance rounds.

Gravity is a tiny force compared to the upward recoil due to the bore line being well above the center of the butt plate. It snaps the barrel like a whip. Floating freely in outer space with no gravity involved, the barrel still rises sharply unless the support is directly behind the barrel. Remember the free pistols that looked like hack saws. Ever wonder why they were made that way? To keep the bore line down in the hand and not above it. MG42 had very little drop in the stock. Guess why. Ever shoot one full speed? Ah! You know why.

Here is a homework assignment. Head to the range at a time when nobody else is there. Go to the bench furthest to the left. Set your favorite schuetzen rifle on on your favorite rest, but on on its side with the barrel down range and the (assumed right-hand) cheek piece up. When you fire the rifle, the recoil will be to the right. Gravity will be down, except perhaps in certain neighborhoods in California. You will then have horizontal vibrations dominant. The vertical component, what there is of it, will not be due to gravity. It will be due to the rest being under the barrel instead of beside it as it would be in a fancier horizontal experiment. 

With the gun in its normal upright position, the mode shapes due to the rest and those due to the recoil are aligned vertically and therefore a bit stronger. Yes, they are in the same direction as gravity, but gravity plays a walk-on role with no lines to speak.

I do not recommend that you reverse the direction of the horizontal experiment if you are a lefty. The lower prong of your schuetzen butt plate will be very near your heart.

An additional point. The barrel does not deform into a single sine wave, but into a series of superimposed waves of specific, related wavelengths. If this were not true, stringed instruments would have no harmonic overtones, and our audio world would be a little less beautiful.