Considering　the　space-periodic　perturbations,　we　prove　the　time-asymptotic　stability　of　the　composite　wave　of　a　viscous　contact　wave　and　two　rarefaction　waves　for　the　Cauchy　problem　of　1-D　compressible　Navier-Stokes　equations　in　this　paper.　Perturbations　of　this　kind　oscillate　in　the　far　field　and　are　not　integrable.　The　key　is　to　construct　a　suitable　ansatz　carrying　the　same　oscillation　as　that　of　the　initial　data,　but　due　to　the　degeneration　of　contact　discontinuity,　the　construction　is　more　subtle.　A　novel　method　for　constructing　robust　ansatzes　is　presented.　It　allows　the　same　weight　function　to　be　used　on　different　variables　and　wave　patterns　while　maintaining　control　over　the　errors.　As　a　result,　it　is　possible　to　apply　this　construction　to　contact　discontinuities　and　composite　waves.　Lastly,　we　demonstrate　that　the　Cauchy　problem　admits　a　unique　global-in-time　solution　and　the　composite　wave　remains　stable　under　space-periodic　perturbations　through　the　energy　method.　(c)　2022　Elsevier　Inc.　All　rights　reserved.