  We found the precise condition for the decay as $t\to\infty$ of Besicovitch
almost periodic entropy solutions of multidimensional scalar conservation laws.
Moreover, in the case of one space variable we establish asymptotic convergence
of the entropy solution to a traveling wave (in the Besicovitch norm). Besides,
the flux function turns out to be affine on the minimal segment containing the
essential range of the limit profile while the speed of the traveling wave
coincides with the slope of the flux function on this segment.
