  For a fixed integer $k$ and a graph $G$, let $\lambda_k(G)$ denote the $k$-th
largest eigenvalue of the adjacency matrix of $G$. In 2017, Tait and Tobin
proved that the maximum $\lambda_1(G)$ among all connected outerplanar graphs
on $n$ vertices is achieved by the fan graph $K_1\vee P_{n-1}$. In this paper,
we consider a similar problem of determining the maximum $\lambda_2$ among all
connected outerplanar graphs on $n$ vertices. For $n$ even and sufficiently
large, we prove that the maximum $\lambda_2$ is uniquely achieved by the graph
$(K_1\vee P_{n/2-1})\!\!-\!\!(K_1\vee P_{n/2-1})$, which is obtained by
connecting two disjoint copies of $(K_1\vee P_{n/2-1})$ through a new edge at
their ends. When $n$ is odd and sufficiently large, the extremal graphs are not
unique. The extremal graphs are those graphs $G$ that contains a cut vertex $u$
such that $G\setminus \{u\}$ is isomorphic to $2(K_1\vee P_{n/2-1})$. We also
determine the maximum $\lambda_2$ among all 2-connected outerplanar graphs and
asymptotically determine the maximum of $\lambda_k(G)$ among all connected
outerplanar graphs for general $k$.
