  We carry out a detailed analysis of a time dispersive dissipative (TDD)
string, using our recently developed conservative and Hamiltonian extensions of
TDD systems. This analysis of the TDD string includes, in particular: (i) an
explicit construction of its conservative Hamiltonian extension, consisting of
the physical string coupled to "hidden strings;" (ii) explicit formulas for
energy and momentum densities in the extended system, providing a transparent
physical picture accounting precisely for the dispersion and dissipation; (iii)
the eigenmodes for the extended string system, which provide an eigenmode
expansion for solutions to the TDD wave equation governing the TDD string. In
particular, we find that in an eigenmode for the extended system the
displacement of the physical string does not satisfy the formal eigenvalue
problem, but rather an equation with a source term resulting from the
excitation of the hidden strings. The obtained results provide a solid basis
for the rigorous treatment of the long standing problem of scattering by a TDD
scatterer, illustrated here by the computation of scattering states for a
string with dissipation restricted to a half line.
