  We perform a systematic study of the evolution of the waveform of black hole
X-ray binary low-frequency QPOs, by measuring the phase difference between
their fundamental and harmonic features. This phase difference has been studied
previously for small number of QPO frequencies in individual sources. Here, we
present a sample study spanning fourteen sources and a wide range of QPO
frequencies. With an automated pipeline, we systematically fit power spectra
and calculate phase differences from archival Rossi X-ray Timing Explorer
(RXTE) observations. We measure well-defined phase differences over a large
range of QPO frequencies for most sources, demonstrating that a QPO for a given
source and frequency has a persistent underlying waveform. This confirms the
validity of recently developed spectral-timing methods performing phase
resolved spectroscopy of the QPO. Furthermore, we evaluate the phase difference
as a function of QPO frequency. For Type-B QPOs, we find that the phase
difference stays constant with frequency for most sources. We propose a simple
jet precession model to explain these constant Type-B QPO phase differences.
The phase difference of the Type-C QPO is not constant but systematically
evolves with QPO frequency, with the resulting relation being similar for a
number of high inclination sources, but more variable for low-inclination
sources. We discuss how the evolving phase difference can naturally arise in
the framework of precession models for the Type-C QPO, by considering the
contributions of a direct and reflected component to the QPO waveform.
