We study the low-energy collective excitations and dynamical response functions of weakly coupled random antiferromagnetic spin-1 / 2 chains. The interchain coupling leads to Néel order at low temperatures. We use the real-space renormalization-group technique to tackle the intrachain couplings and treat the interchain couplings within the random phase approximation ͑RPA͒. We show that the system supports collective spin wave excitations, and calculate the spin wave velocity and spectra weight within RPA. Comparisons will be made with inelastic neutron scattering experiments on quasi-one-dimensional disordered spin systems such as doped CuGeO 3 . DOI: 10.1103/PhysRevB.72.020403 PACS number͑s͒: 75.10.Jm, 75.30.Cr, 75.30.Ds, 75.50.Ee Antiferromagnetic ͑AF͒ quantum spin chains have been of interest to physicists since the early days of quantum mechanics. 1 The one-dimensional nature of such systems allowed for tremendous theoretical progress both in clean systems by using exact solution and field theory mapping 1,2 and disordered systems within renormalization-group framework. 3-8 While such one-dimensional models have remarkably rich physics, in general, they do not give a complete description of real systems. Real spin chain compounds, such as CuGeO 3 ͑Ref. 9͒ and KCuF 3 , 10 always have some weak interchain couplings present, which can change the physics at lowest energy and/or temperature. For example, strictly one-dimensional ͑1D͒ models do not exhibit phase transitions into states with broken symmetry, while real spin chain systems often develop Néel order at very low temperatures due to the weak ͑3D͒ interchain couplings. It is thus important to study the effects of these interchain couplings to fully understand the low-energy and/or temperature physics of real spin chain compounds.In this paper we study the low-energy collective excitations and dynamical response functions of weakly coupled, disordered AF spin-1 / 2 chains. Our work is motivated in part by the experimental studies on doped CuGeO 3 . In the absence of doping, it is a spin-Peierls system in which the spins dimerize and form a gapped, nonmagnetic ground state. Upon doping, the system becomes disordered, and both dimerization and spin gap get suppressed. Amazingly, when doping reaches certain levels the spins become Néel ordered at low temperature, which has been observed experimentally in Zn-and Si-doped CuGeO 3 . [11][12][13][14][15][16][17][18][19][20] Since these experimental discoveries a number of theoretical papers have addressed the static Néel ordering in these systems 24-36 using meanfield theory. On the other hand the collective excitations and dynamical response functions which have been studied experimentally using inelastic neutron scattering, 20 have not been studied theoretically thus far. The collective excitations and dynamical response functions are the subjects of the present work. We go beyond mean-field theory by allowing the Néel order parameter to fluctuate, and treat the interchain coupling using the random-phase a...