  Binary-black-hole orbits precess when the black-hole spins are mis-aligned
with the binary's orbital angular momentum. The apparently complicated dynamics
can in most cases be described as simple precession of the orbital angular
momentum about an approximately fixed total angular momentum. However, the
imprint of the precession on the observed gravitational-wave signal is yet more
complicated, with a non-trivial time-varying dependence on black-hole dynamics,
the binary's orientation and the detector polarization. As a result, it is
difficult to predict under which conditions precession effects are measurable
in gravitational-wave observations, and their impact on both signal detection
and source characterization. We show that the observed waveform can be
simplified by decomposing it as a power series in a new precession parameter $b
= \tan(\beta/2)$, where $\beta$ is the opening angle between the orbital and
total angular momenta. The power series is made up of five harmonics, with
frequencies that differ by the binary's precession frequency, and individually
do not exhibit amplitude and phase modulations. In many cases, the waveform can
be well approximated by the two leading harmonics. In this approximation we are
able to obtain a simple picture of precession as caused by the beating of two
waveforms of similar frequency. This enables us to identify regions of the
parameter space where precession is likely to have an observable effect on the
waveform, and to propose a new approach to searching for signals from
precessing binaries, based upon the two-harmonic approximation.
