Congrats on the osage Jack,
My take on it is this.
I have made lots of self bows from greenwood. Meaning that It was cut and with in 6 months I made the bow.
This is how that process would work,
Take fresh cut log and recuce it to bow blank dimensions. to the floor tiller stage.
SEAL it with some heavy duty sealer. Then take that "bow blank" and steam it and straighten it on your caul. This will drive out the "green" moisture. Now it is very important to let the newly "seasoned" bow blank to re-hydrate.
I always left mine alone for a month at least. Michigan is wet in the summer and dry (in my shop/house) in the winter so use your judgment here.
This will make a nice bow. One of my favorites was cut in Feb and shooting in May.
I re-worked the handle on that bow a little this past fall and hunted with it. I first made the bow in 2002 and shot 100's and 1000's of arrows thru it until I became corupt with the glass bows. That bow still hold 1.5" of reflex.
Gary Davis goes thru this in some detail in his bow making DVD.
http://www.3riversarchery.com/Rattlestick%3A+A+Selfbow+Tutorial+DVD+By+Gary+S.+Davis_i7041_baseitem.html I still feel that a better bow can be made from good air dried wood. 3-5yrs.
Merry Christmas and Happy New Year.
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