Microwave　pulses　are　used　ubiquitously　to　control　and　measure　qubits　fabricated　on　superconducting　circuits.　Due　to　continual　environmental　coupling,　qubits　undergo　decoherence　either　when　it　is　free　or　when　it　is　coupled　to　an　incident　pulse.　We　study　theoretically　the　decoherence-induced　effects　when　a　qubit　is　subject　to　the　driving　of　time-dependent　pulses,　which　can　accomplish　geometric　logic　gating,　under　a　dissipative　environment　with　linear　spectral　distribution.　We　find　that　a　transmissible　pulse　of　finite　width　adopts　an　asymmetric　multi-hump　shape,　due　to　the　imbalanced　pumping　and　emitting　rates　of　the　qubit　during　inversion　when　the　environment　is　present.　The　pulse　shape　reduces　to　a　solitonic　pulse　at　vanishing　dissipation　and　a　pulse　train　at　strong　dissipation.　We　give　detailed　analysis　of　the　environmental　origin　from　both　the　perspectives　of　envelope　and　phase　of　the　propagating　pulse.