The broad line wings in molecular cloud spectra are proposed to result from strong magnetic waves on the periphery of dense cores and in the intercore regions where the Alfvén velocity should be larger than average. The observed line profiles are reproduced by a simple but realistic model, and the ratio of the broad to the narrow line components is found to equal approximately three, independent of cloud parameters, as long as the core/intercore contrast in the local average density is sufficiently large. Interactions between the magnetic waves should produce dense clumps in the non-linear splash regions between converging flows.