  Direct numerical simulations (DNS) of turbulent channel flows up to
$Re_{\tau} \approx 1000$ are conducted to investigate the three-dimensional
(consisting of streamwise wavenumber, spanwise wavenumber and frequency)
spectrum of wall pressure fluctuations. To develop a predictive model of the
wavenumber-frequency spectrum from the wavenumber spectrum, the time
decorrelation mechanisms of wall pressure fluctuations are investigated. It is
discovered that the energy-containing part of the wavenumber-frequency spectrum
of wall pressure fluctuations can be well predicted using a similar random
sweeping model for streamwise velocity fluctuations. To refine the
investigation, we further decompose the spectrum of the total wall pressure
fluctuations into the auto spectra of rapid and slow pressure fluctuations, and
the cross spectrum between them. We focus on evaluating the assumption applied
in many predictive models, that is, the magnitude of the cross spectrum is
negligibly small. The present DNS shows that neglecting the cross spectrum
causes a maximum error up to 4.7dB in the sub-convective region for all
Reynolds numbers under test. Our analyses indicate that the assumption of
neglecting the cross spectrum needs to be applied carefully in the
investigations of acoustics at low Mach numbers, in which the sub-convective
components of wall pressure fluctuations make important contributions to the
radiated acoustic power.
