  We reconsider the problem of the enhancement of tunnelling of a quantum
particle induced by disorder of a one-dimensional tunnel barrier of length $L$,
using two different approximate analytic solutions of the invariant imbedding
equations of wave propagation for weak disorder. The two solutions are
complementary for the detailed understanding of important aspects of numerical
results on disorder-enhanced tunnelling obtained recently by Kim et al. (Phys.
rev. B{\bf 77}, 024203 (2008)). In particular, we derive analytically the
scaled wavenumber $(kL)$-threshold where disorder-enhanced tunnelling of an
incident electron first occurs, as well as the rate of variation of the
transmittance in the limit of vanishing disorder. Both quantities are in good
agreement with the numerical results of Kim et al. Our non-perturbative
solution of the invariant imbedding equations allows us to show that the
disorder enhances both the mean conductance and the mean resistance of the
barrier.
