Summary: Consider the transmission eigenvalue problem for \(u \in H^1 (\Omega)\) and \(v \in H^1 (\Omega)\): \[ \begin{aligned} \begin{cases} \nabla \cdot (\sigma \nabla u) + k^2 \mathbf{n}^2 u = 0 \quad &\text{in } \Omega,\\ \Delta v + k^2 v = 0 \quad &\text{in } \Omega,\\ u = v, \sigma \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} &\text{on } \partial \Omega, \end{cases} \end{aligned} \] where \(\Omega\) is a ball in \(\mathbb{R}^N\), \(N = 2, 3\). If \(\sigma\) and \(\mathbf{n}\) are both radially symmetric, namely they are functions of the radial parameter \(r\) only, we show that there exists a sequence of transmission eigenfunctions \(\{u_m, v_m\}_{m \in \mathbb{N}}\) associated with \(k_m \rightarrow + \infty\) as \(m \rightarrow + \infty\) such that the \(L^2\)-energies of \(v_m\)’s are concentrated around \(\partial \Omega\). If \(\sigma\) and \(\mathbf{n}\) are both constant, we show the existence of transmission eigenfunctions \(\{u_j, v_j\}_{j \in \mathbb{N}}\) such that both \(u_j\) and \(v_j\) are localized around \(\partial \Omega\). Our results extend the recent studies in [Y. T. Chow et al., SIAM J. Imaging Sci. 14, No. 3, 946–975 (2021; Zbl 1478.35159)]. Through numerics, we also discuss the effects of the medium parameters, namely \(\sigma\) and \(\mathbf{n}\), on the geometric patterns of the transmission eigenfunctions.