  In this work we consider a wide range of energy critical wave equation in
3-dimensional space with radial data. We are interested in exterior scattering
phenomenon, in which the asymptotic behaviour of a solutions $u$ to the
non-linear wave equation is similar to that of a linear free wave $v_L$ in an
exterior region $\{x: |x|>R+|t|\}$, i.e.
  \[
  \lim_{t\rightarrow \pm \infty} \int_{|x|>R+|t|} (|\nabla(u-v_L)|^2 +
|u_t-\partial_t v_L|^2) dx = 0.
  \]
  We classify all such solutions for a given linear free wave $v_L$ in this
work. We also give some applications of our theory on the global behaviours of
radial solutions to this kind of equations. In particular we show the
scattering of all finite-energy radial solutions to the defocusing energy
critical wave equations.
