But in fact the terminal impedance of a lossless semi-infinite
transmission line is purely real.  There is no reactive component.
In an electrical circuit it behaves just like a resistor in that it
absorbs energy.  That is not inconsistent with energy conservation,
however, because the energy that the transmission line absorbs is
stored in the electromagnetic fields within the transmission line.
If a voltage or current of finite duration is applied at the terminals
of the line the fields will take the form of travelling waves which
propagate away from the terminals.  Because the line is semi-infinite
the travelling wave does not reflect back toward the terminals.  That
is not true, by the way, for a line of finite length.  The latter does
have a purely imaginary terminal impedance.

This is not so different from an ordinary resistor which dissipates
the energy it absorbs as heat.  In that case the absorbed energy is
stored as the motion of atoms or molecules in the environment.  Perhaps
an even better analogy is free space which absorbs any electromagentic
energy that is radiated by any source (e.g., an atom decaying from an
excited state to its ground state).

As I said in a previous post, I highly recommend analyzing the damped
LC circuit by substituting a transmission line for the damping resistor
and writing down the resultinh Lagrangian or Hamiltonian.  This simple
model is exactly solvable and will teach you a great deal about the
physics of dissipative systems.

Mike