The Double Haul is an essential part of modern fly casting. It is very difficult to achieve great distance without it, and you will certainly cast far greater distance with it.
The science of it is in loading (bending the rod). As you move the rod forwards or backward with your casting arm you give a sharp pull with your other hand rather than leaving your hand steady. The more the rod bends and the harder you haul, the faster the line speed. By hauling, you are basically accelerating the line forward and backwards faster than if you were not hauling. The faster the rod tip and hence forth the line moves through the air, then the more line that will shoot once you stop the rod.
In the beginning I have notice that people will generally pull down on the back cast and then undo the good work by moving the hauling hand back to its start position on the forward cast. This has the effect of loading on the back-cast - excellent, but unloading on the front cast - terrible. What you actually want to be doing on the back cast is pull-return. And on the front cast pull-return.
A common difficulty is that when the line is short people will pull to hard and unfortunately you won't be able to return to the start position with your hauling hand because the slack won't be taken up i.e. it won't shoot enough. What you need to remember is "short line, short haul" i.e. when you start the cast just give a little tiny tug, and as you get more line aerialised then haul longer and harder and before you know it your hauling hand will nearly be meeting your casting hand at the end of each forward and backwards stroke.
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