  We study the coupling of bouncing-ball modes to chaotic modes in
two-dimensional billiards with two parallel boundary segments. Analytically, we
predict the corresponding decay rates using the fictitious integrable system
approach. Agreement with numerically determined rates is found for the stadium
and the cosine billiard. We use this result to predict the asymptotic behavior
of the counting function N_bb(E) ~ E^\delta. For the stadium billiard we find
agreement with the previous result \delta = 3/4. For the cosine billiard we
derive \delta = 5/8, which is confirmed numerically and is well below the
previously predicted upper bound \delta=9/10.
