  In this paper we will study certain models of irreducible admissible
representations of the split special orthogonal group $SO(2n+1)$ over a
nonarchimedean local field. If $n=1$, these models were considered by
Waldspurger. If $n=2$, they were considered by Novodvorsky and
Piatetski-Shapiro \cite{NP}, who called them {\it Bessel models}. They arise
from a variety of Rankin-Selberg integrals nad the resultos of this paper will
naturally have applications to the study of L-functions. They also arise in the
study of the theta correspondence between $SO(2n+1)$ and the double cover of
$Sp(2n)$, and they will therefore be of importance in generalizing the work of
Waldspurger. As a global application we consider the Eisenstein series on
$SO(2n+1)$ formed with a cuspidal automorphic representation $\pi$ on $GL(n)$,
and we show that its Bessel period (6.2) is essentially a product of L-series.
This generalizes work of B\"ocherer and Mizumoto.
