The attached eddy hypothesis of Townsend (The structure of turbulent shear flow, 1956,Cambridge U. Press) states that the logarithmic mean velocity would admit self-similarenergy-containing eddies which scales with the distance from the wall. Over the pastdecade, there has been significant amount of evidence supporting the hypothesis, placingit to be the central platform for the statistical description of the general organisationof coherent structures in wall-bounded turbulent shear flows. Nevertheless, the mostfundamental question, namely why the hypothesis has to be true, remains unansweredover many decades. Under the assumption that the integral length scale is proportionalto the distance from the wally, in the present study, we analytically demonstrate thatthe mean velocity is a logarithmic function ofyif and only if the energy balance atintegral length scale is self-similar with respect toy, providing a theoretical groundfor the attached eddy hypothesis. The analysis is subsequently verified with the datafrom direct numerical simulation of incompressible channel flow at the friction ReynoldsnumberReτ'5200 (Lee & Moser,J. Fluid Mech., vol. 774, 2015, 395–415).