  Consider the transmission eigenvalue problem for $u \in H^1(\Omega)$ and
$v\in H^1(\Omega)$ associated with $(\Omega; \sigma, \mathbf{n}^2)$, 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 [15,16]. Through
numerics, we also discuss the effects of the medium parameters, namely $\sigma$
and $\mathbf{n}$, on the geometric patterns of the transmission eigenfunctions.
