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Recent observational and modeling evidence has demonstrated that planetary waves can modulate atmospheric tides, and secondary waves arising from their nonlinear interactions are an important source of both temporal and longitude variability in the thermosphere. While significant progress has been made on understanding how this form of vertical coupling occurs, uncertainty still exists on how the horizontal structures of primary and secondary waves evolve with height and the processes responsible for this evolution, in part due to lack of global observations between 120 km and 260 km. In this work we employ a Thermosphere Ionosphere Mesosphere Electrodynamics general circulation model simulation covering all of 2009 that is forced by Modern-Era Retrospective Analysis for Research and Applications dynamical fields, to assess the relative contribution of zonal mean winds and molecular dissipation on the vertical coupling of the eastward propagating diurnal tide with zonal wave number 3 (DE3), the 3 day ultrafast Kelvin wave, and the secondary waves arising from their nonlinear interaction. By developing and applying a new analytic formulation describing the latitudinal structure of an equatorially trapped wave subject to dissipation and background winds, we show that dissipation is the primary contributor to the broadening of the latitudinal structures with height, while asymmetries in the background wind field are responsible for the distortion of the height-latitude structures. [ABSTRACT FROM AUTHOR]