  The characteristics of the wavenumber-frequency spectra of the rapid, slow
and total wall pressure fluctuations are investigated using direct numerical
simulation (DNS) of turbulent channel flow up to $\Rey_\tau\approx 1000$. For
the wavenumber-frequency spectra of the total wall pressure fluctuations, a
valley-like behavior of contour lines in the sub-convective region is found,
which may be linked to the Kraichnan-Phillips theorem. For the decomposition of
the wall pressure spectra, it is commonly assumed in previous studies that the
cross spectral density (CSD) between the rapid and slow components of the wall
pressure fluctuations can be neglected. Yet no experimental or numerical
evidence is available for either confirming or disproving this assumption. In
this paper, we use DNS data to quantitatively evaluate this assumption.
Emphasizes are put on the error in decibel scale caused by neglecting the CSD
between the rapid and slow components of the wall pressure fluctuations. It is
found that this assumption is approximately accurate for one- and
two-dimensional spectra, but causes a large magnitude of error in the
three-dimensional wavenumber-frequency spectra. An error of 5dB is observed in
the sub-convective region and such a large error is observed for a wide range
of Reynolds numbers ($180\le\Rey_\tau\le 1000$). The analyses show that the
assumption of neglecting the CSD needs to be applied carefully at the scales
falling in the sub-convective region.
