When an arrow is released using fingers/tab the nock end of the arrow initially bends away from the bow. The nock end subsequently bends back towards the bow and then away again. Archer's paradox is the term used to describe this bending behavior. The aim is to have the nock (fletching) end of the arrow bending away from the bow as it passes the body of the bow the purpose being to avoid any collision between the rear of the arrow and the bow. This section gives an overview of how this bending behavior is generated. I will assume a right handed archer when describing directions of travel etc.

Points to bear in mind are that the force exerted by the string on the arrow decreases as the arrow moves forward and also that the direction of the string force is always towards the string bracing height position.
 

THE LOOSE - NOCK END

The pressure button is set up for a recurve bow so that the arrow at full draw is pointing left away from the bow. The direction of the string force at full draw points towards the bracing height position and is therefore on the bow side of the arrow. If the loose was made using a mechanical release then the nock end of the arrow would bend towards the bow and the pile end of the arrow would rotate away from the bow as illustrated. Ooops! The arrow would end up flying uncontrollably off to the left.

What initializes the Archer's Paradox effect is the action of the fingers/tab on the bow string. At full draw the string force is balanced by an equal and opposite force on the tab. At the loose the string force causes the tab to rotate as the fingers are 'uncurled'. At this point there are three forces acting, the string force towards the bracing height position, the tab reaction force at a right angle to the tab surface and a tab frictional force parallel to the tab surface. These three forces add together  to produce a net force on the string forwards and to the left away from the bow. The sideways acceleration of the string causes the arrow shaft to bend away from the bow at the nock and as a consequence the string force ends up running across the arrow shaft. At the same time the forwards acceleration of the string transfers the load from the tab onto the nock end of the arrow shaft. The purpose in having a 'slippy' tab is that the lower the tab frictional force then the shorter is the time that the string is acting on the tab during the loose i.e. there is less time to mess it up by moving the string hand. The other effect the tab has in principle is on the effective arrow dynamic spine. The 'slippier' the tab then the higher the nock acceleration will be and the more the arrow will bend.

The result is that when the string force ends up acting on the arrow nock the string force runs across the arrow shaft and the reaction at the nock (which is at a right angle to its surface) is pointing backwards and away from the bow to the left. These two forces add together and the resultant force direction is forwards and to the left away from the bow i.e. the nock end of the arrow continues to bend away from the bow. This process is similar to the behavior of a pole vaulter's pole after it is planted in the box.

THE LOOSE - PILE END

As the nock end of the arrow bends away from the bow a torque is generated on the arrow in the direction to rotate the arrow pile towards the bow. The arrow ends up leaning on the side of the bow or pressure button as appropriate. Because the arrow is pushing against the bow there is an opposite reaction with the bow/button applying a sideways force on the arrow. This force causes the arrow to bend in the direction for the pile to move away from the bow.

THE STRAIGHTEN

If the nock end of the arrow continued to bend away from the bow then eventually the arrow would snap. What limits the amount of this arrow bending? The best way to look at this is to regard the arrow as a spring with a weight on the front end being pushed from the rear. When the arrow is released the nock end of the arrow accelerates forward faster than the pile end which is heavy and has high inertia. The gap between pile and nock reduces and this reduction goes into spring compression of the arrow (it bends). The acceleration of the pile comes from shaft forces on it which comprise the string force through the shaft and also from this spring compression  (bending) of the shaft. As the arrow bends more the pile acceleration keeps increasing until it exceeds the nock acceleration. At some point the pile forward speed catches up with the nock forward speed at which point the arrow stops bending. The pile is still accelerating faster than the nock (string force + spring force) so the pile now travels faster than the nock and the spring compression comes out of the arrow i.e. it straightens up. As the rear of the arrow straightens the force with which the pile end of the arrow is pressing against bow/button from torque reduces and so the bend in the front part of the shaft reduces. As the arrow straightens the pile acceleration decreases. You end up with a more or less straight arrow with the pile and nock traveling forwards at the same speed.

The differences between this 'straight' arrow and the one at full draw is that in addition to the arrow being further forward and so the string force and direction to the brace height position being different the arrow shaft has a lateral velocity profile. Not all the spring energy goes into pile acceleration, much of it has gone into a lateral shaft acceleration.

THE SECOND BEND

If at this point the movement of the string was somehow frozen then the arrow would end up oscillating as described in the section on arrow vibration. This vibration happens and itself triggers a nock end bending effect similar to the loose. When the shaft has straightened up it then, because the shaft is moving sideways, bends outwards. The net effect of the string force across the shaft and the reaction force of the nock produce a net force accelerating the arrow nock end forwards and laterally towards the bow. The nock end of the arrow bends towards the bow as the arrow travels forward. You have exactly the same pile/nock forward speed shuffle as before for the arrow to reach a maximum bend and then straighten up again. There is no bow/button to counteract the torque from the rear end bending this time and the torque results partly in shaft rotation and partly in making the front part of the shaft bend. The swinging of the front section of the shaft provides a counterbalancing torque to the rear end bending. (This was a trick used by dinosaurs who used a long tail with a lump of bone on the end as a weapon. When the tail was swung the long neck/head was swung in the opposite direction, counterbalancing the torque so the dingo was not spun off its feet).

THE THIRD BEND

Following the arrow straightening from the second bend the whole process repeats with the nock end of the arrow bending away from the bow on its third bend and so on as long as the string is accelerating the arrow.

During this process, at around the point the arrow completes its second bend  the arrow leaves the string.. The aim is to have a clean separation of the string from the nock and to have the rear end of the arrow sufficiently bent away from the bow to provide good clearance as it passes the riser. Having the right timing to do this relates to all the factors which affect how much and how rapidly the arrow bends and how fast the arrow accelerates forward. i.e. arrow length, mass, shaft spine, pile weight, draw weight, bracing height and the bow force draw curve. Fortunately all the archer needs to do all this is to select the correct arrow from a selection chart based on accumulated experience of what works and what does not.

The amount the arrow bends ("weak/stiff") depends on the relative accelerations of the nock and pile ends. Increasing the pile weight for example reduces the pile acceleration and hence the arrow bends more ("weaker") and vice versa. You can similarly change the amount the arrow bends (its "stiffness") by changing the weight (and hence acceleration) at the nock end of the arrow. As the nock end of the arrow is light the arrow stiffness is fairly sensitive to changes in nock weight. e.g. adding brass nocking points or "pin nocks" reduces the nock acceleration and therefore stiffens the arrow. A special case is the fletchings. These increase the nock weight but in addition the drag on the fletching surface additionally reduces the nock acceleration. Fitting larger area fletchings (with the same weight) will stiffen the arrow due to the increased drag.