The signed oak bow is obtained during the Ghosts Ahoy quest. It is an oak longbow signed by Robin in Port Phasmatys after the player beats him in several games of Rune-Draw.
Acquiring and Using the Bow
Talk to Ak-Haranu. He has the translation manual you need for the quest, but wants an oak longbow signed by Robin, the master bowman, in exchange for it. Luckily, he is residing in the nearby inn. The player then has to go to Green Ghost Inn. Talk to Robin, and he will refuse to sign the bow, but offer to play a game of Rune-Draw. Win four rounds, and he will sign the bow to repay the debt. Go back to Ak-Haranu and give him the bow and he will give you the manual.