2. As shown in Figure $1, O$ is the center of the regular octagon $A B C D E F G H$, and $X$ is the midpoint of side $A B$. Then the area of the shaded part $S_{1}$ occupies the total area $S$ ( ).
(A) $\frac{11}{32}$
(B) $\frac{3}{8}$
(C) $\frac{13}{32}$
(D) $\frac{7}{16}$
(E) $\frac{15}{32}$
2. D.

From the given, $S_{\triangle O B X}=\frac{1}{2} S_{\triangle A O B}$.
Since $S_{\triangle A O B}=S_{\triangle B O C}=S_{\triangle C O D}=S_{\triangle D O E}$, therefore,
$$
\frac{S_{1}}{S}=\frac{3 S_{\triangle A O B}+\frac{1}{2} S_{\triangle A O B}}{8 S_{\triangle A O B}}=\frac{7}{16}
$$