  We reexamine here a problem considered in detail before by Waugh and Lake:
the solutions of spherically symmetric Einstein's equations with a radial flow
of unpolarized radiation (the Vaidya metric) in double-null coordinates. This
problem is known to be not analytically solvable, the only known explicit
solutions correspond to the constant mass case (Schwarzschild solution in
Kruskal-Szekeres form) and the linear and exponential mass functions originally
discovered by Waugh and Lake. We present here a semi-analytical approach that
can be used to discuss some qualitative and quantitative aspects of the Vaidya
metric in double-null coordinates for generic mass functions. We present also a
new analytical solution corresponding to $(1/v)$-mass function.
