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The recently discovered coexistence of incommensurate antiferromagneticneutron scattering peaks and commensurate resonance in underdopedYBa$_2$Cu$_3$O$_{6+x}$ is calling for an explanation. Within the t-J model, thedoping and energy dependence of the spin dynamics of the underdoped bilayercuprates in the normal state is studied based on the fermion-spin theory byconsidering the bilayer interactions. Incommensurate peaks are found at$[(1\pm\delta)\pi,\pi] $ and $[\pi,(1\pm\delta)\pi]$ at low energies with$\delta$ initially increasing with doping at low dopings and then saturating athigher dopings. These incommensurate peaks are suppressed, and the parameter$\delta$ is reduced with increasing energy. Eventually it converges to the$[\pi,\pi]$ resonance peak. Thus the recently observed coexistence isinterpreted in terms of bilayer interactions.Comment: 15 pages, Revtex, five figures are included, accepted for publication  in Phys. Rev.

Convergence of energy-dependent incommensurate antiferromagnetic neutron scattering peaks to commensurate resonance in underdoped bilayer cuprates