  This paper estimates the location and the width of the nodal set of the first
Neumann eigenfunctions on a smooth convex domain $\Omega \subset \mathbb R^n$,
whose length is normalized to be 1 and whose cross-section is contained in a
ball of radius $\epsilon$. In \cite{CJK2009}, an $O(\epsilon)$ bound was
obtained by constructing a coordinate system. In this paper, we present a
simpler method that does not require such a coordinate system. Moreover, in the
special case $n = 2$, we obtain an $O(\epsilon^2)$ bound on the width of the
nodal set, in analogy to the corresponding result in the Dirichlet case
obtained in \cite{GJ1995}.
