In rough-wall boundary layers, wall-parallel non-homogeneous mean-flow solutions exist that lead to so-called dispersive velocity components and dispersive stresses. They play a significant role in the mean-flow momentum balance near the wall, but typically disappear in the outer layer. A theoretical framework is presented to study the decay of dispersive motions in the outer layer. To this end, the problem is formulated in Fourier space, and a set of governing ordinary differential equations per mode in wavenumber space is derived by linearizing the Reynolds-averaged Navier–Stokes equations around a constant background velocity. With further simplifications, analytically tractable solutions are found consisting of linear combinations of
$\exp (-kz)$
and
$\exp (-Kz)$
, with
$z$
the wall distance,
$k$
the magnitude of the horizontal wavevector
$\boldsymbol{k}$
, and where
$K(\boldsymbol{k},Re)$
is a function of
$\boldsymbol{k}$
and the Reynolds number
$Re$
. Moreover, for
$k\rightarrow \infty$
or
$k_{1}\rightarrow 0$
(with
$k_{1}$
the stream-wise wavenumber),
$K\rightarrow k$
is found, in which case solutions consist of a linear combination of
$\exp (-kz)$
and
$z\exp (-kz)$
, and are independent of the Reynolds number. These analytical relations are compared in the limit of
$k_{1}=0$
to the rough boundary layer experiments by Vanderwel & Ganapathisubramani (J. Fluid Mech., vol. 774, 2015, R2) and are in reasonable agreement for
$\ell _{k}/\unicode[STIX]{x1D6FF}\leqslant 0.5$
, with
$\unicode[STIX]{x1D6FF}$
the boundary-layer thickness and
$\ell _{k}=2\unicode[STIX]{x03C0}/k$
.