buffer layer using the the Kolmogorov-Obukhov energy spectrum of turbulent fluctuations. The form of the MVP in the wake is also obtained. This was done by summing the energy of the wall-attached eddies, as hypothesised originally by Townsend in [@T76].

In this paper, we propose a generalisation of the spectral theory that includes fluctuations in the streamwise velocity due to an essentially fluctuating wall shear stress. Fig. \[fig:wx1\] shows the spectral theory predictions of the profiles of the mean velocity and mean square velocity fluctuations across the viscous, buffer and inertial layers. The rest of the paper is structured as follow. We summarise the analysis in Ref. [@GGGC10] and its extension in Section \[sec:MVP\], and generalise it to include the fluctuations in Section \[sec:fluct\]. This produces the log law of the wall in Eq. (\[eq:l-lfluct\]) for the velocity fluctuations and its higher moments in Eq. (\[eq:l-pfluct\]). Then in Section \[sec:functional\], we derive the functional form of the mean-square fluctuations in the viscous layer and the inertial layer. In Section \[sec:SCT\], we use the attached eddy hypothesis and the stochastic closure theory [@BB211; @BB314] to derive the form of the Townsend-Perry and the generalized Townsend-Perry constants. This allows us to derive 