Spin density waves (SDWs) represent a fundamental paradigm of spatially modulated order in condensed matter systems, yet their electrical and mechanical analogs—polarization and strain density waves (PDWs and StDWs)—have remained elusive as equilibrium phases. Here, we introduce a general, symmetry-driven strategy to unlock static PDWs and StDWs in perovskites ${\mathrm{SrTiO}}_3$ and ${\mathrm{SrMnO}}_3$. Using first-principles calculations, we uncover a previously overlooked soft antiferrodistortive tilt gradient mode at small-$q$ wave vector in the phonon dispersion of their presumed $Ima2$ ground state under moderate tensile strain. Group-theory analysis reveals that a hard polar-acoustic phonon, which intrinsically carries PDWs and StDWs, is improperly destabilized by a trilinear coupling with this modulated tilt mode and an inherently uniform tilt mode. This interaction drives a structural transition from the $Ima2$ phase to a novel lower-energy $Pmn{2}_1$ phase that hosts long-range-ordered PDWs and StDWs. Strikingly, the engineered StDWs in ${\mathrm{SrMnO}}_3$ activate an electrically tunable SDW via the flexomagnetic effect. These discoveries fundamentally revise the strain-phase diagrams of prototypical perovskites and establish a unified phonon engineering framework that links modulated phonon instabilities to targeted density-wave order, offering new pathways for designing advanced electromechanical and magnetoelectric functionalities.

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• Elasticity
• Ferroelectricity
• Magnetism
• Phase diagrams
• Phase transitions
• First-principles calculations