This might help. First I entered the query below.
y=exp(-x/5)*cos(2*pi*x) between y=-.4 and y=.4
This gave a picture with the graph clipped between -.4 and .4. Not exactly what was wanted, but I could see visually that the clipping was no longer in effect shortly before x=5. So I then entered this next query.
y=exp(-x/5)*cos(2*pi*x) between x=4 and x=6
If I understand the problem then the result from this second query might include what you need.
By the way, the enveloping values (local extrema) are determined from just the exponential part. So possibly you just want to know when the oscillation extrema fall in that bound independent of where it started (that is, independent of phase, which here is assumed to start at a peak at time 0). For this you just enter as below.
.4=exp(-x/5)
Wolfram|Alpha then tells you this happens around 4=4.58.