The theory for waves on a buoyant fluid conduit in a more viscous oute
r fluid is extended to include a visco-elastic outer fluid. The extern
al fluid is treated as a linear Kelvin-type visco-elastic medium and a
wave evolution equation is derived. This equation is identical to the
purely viscous case with the exception of a new term representing the
elastic effects. A conservation law is derived and used in an analyti
c treatment for a slowly-varying solitary wave (given initially by the
exact solution to the purely viscous case) for the case of small, but
non-zero, elasticity. The theory shows that the wave amplitude will d
ecay and a shelf. required for the conservation of mass, will develop
behind the wave. Numerical solutions of the evolution equation support
the analytic approximation. Laboratory experiments show qualitative a
greement with the analytic and numerical development. Geophysical appl
ications suggest that these effects may be most important for melt mig
ration in the asthenosphere.