Energy transfer through a longbow "Dry firing", whereby a bowstring is pulled and released without an arrow is meant to be a big no-no. I can understand this for compound bows, where there are pulleys involved. However,  for a longbow and given the mass of an arrow is so small, how would the energy transfer caused by such an action, of firing without an arrow to cause damage to the bow, allegedly catastrophically.  We must assume the bow is sound with no weakness.
 A: It will depend quite strongly on the specifics of the bow. Some bows won't care, some will be seriously harmed.
The point is, the string is very much lighter than the arrow, light as the arrow is. With no arrow, the string can get to the "un-pulled" position very much faster. This means the part of the bow that provides the tension can have a short period of nearly zero resistance. That means it can spring back very much faster than the usual way it does when there is an arrow. And then it eventually hits the full strength of the string. That means the energy goes into motion of the bow parts instead of the arrow, and then gets hard-stopped on the string. So this can provide a substantial shock to the bow, very much larger than the usual case with an arrow. Nearly the entire energy load of the bow will wind up in this shock, and it will be applied over a very short time.
Some bow designs won't be particularly harmed by this. For example, in some bow designs, the tension-providing parts of the bow move very little. Possibly the string will take the worst of it. If the string is a little heavier, as it is in some designs, then the relative speed of the string with no arrow will be reduced.
It seems that the problem is dramatically more pronounced in compound bows than in regular old "simple arc" bows. This web site has several vids of compound bows failing catastrophically. Some of these cases make me wonder why there are not more bow-hunting injuries reported.
http://anchorthatpoint.com/dry-firing-a-compound-bow/
This guy discusses dry firing in the context of several types of bow.
https://www.youtube.com/watch?v=gPII-cFUKzY
