Each detail is a function of time. Its length is the same as the
Each detail represents the frequency content of the band between the
detail before and the detail after.
But I think it's easier to think in terms of "timescale", not
frequency, where timescale = 1/frequency.
The timescales double with each detail.
In my experience with many real datasets, the timescale of the 1st
detail is 3*dt, where dt is the interval between data, though the
textbooks say it is 2*dt. I've come to this by doing zero-crossing
analyses on details from all sorts of data. Thus, in practice, I've
found the timescales go 3, 6, 12, 24, 48, 96, ...*dt, never mind what
the mathematicians say.
One other point about time information.
If you take the variance of each detail and plot it against timescale,
you will get a spectrum of sorts. It's not the same as a Fourier
spectrum because the intervals between data are not equal in
frequency, so there's a problem of how to scale the thing. However,
if you sum the variances, you'll get the variance of the original
signal, just as with taking the area under a Fourier spectrum. Now,
when you take the variance of the details, you're removing the time
dependence, just as a Fourier spectrum does. And that's the
difference. Wavelet details have their energy distributed with time.