a formula for $\tilde y$ using this equation along with the equation $${\kappa}^2 {\tilde y}^2( u')^2 I^{3/4}+u' = \frac{\tau_0}{\langle \tau_0 \rangle}\left(1-\frac{y}{Re\sqrt{f}}\right).$$ The resulting formula is given in Ref. [@GGGC10], $$\tilde y = \left(\frac{(\eta/y)^{4/3}+ \kappa^{4/3}I^{1/2}(\eta/y,0)}{\kappa^{2/3}(\eta/y)^{8/3}I^{1/4}(\eta/y,0)}\right).$$ It gives the minimum value of $\tilde y$ for which $I(\eta/y,0)>0$ and the small eddies begin to contribute to the turbulent shear stress $\tau_t >0$. In fact for each value of the parameter $\beta_d$ there is a minimum value of $\tilde y$ denoted $\tilde y_v$ below which $I=0$. Only after this minimum does $\tilde y$ increase with $\eta/y$. This gives the end of the viscous layer and the beginning of the buffer layer and a value of the MVP, $u_v$ at $\tilde y_v$. It also gives the value of the fluctuation $w$ at $\tilde y_v$ and we can integrate the differential equations for $u$ and $w$, with respect to $y$, to get the form of both functions in the buffer layer, inertial layer and the wake. Along with the formulas in the viscous layer this gives the full functional form. The differential equations use the spectral information through the full functional form of $I$ and the two parameters $\beta_d$ and $\beta_e$ must be fitted to experimental 