Buckling of a synthetic film ($\kappa q^4+\sigma q^2<0$)

A flat, unsupported film's bending and tension energies give modal stiffness $\kappa q^4+\sigma q^2$. Negative tension represents compression, destabilizing sufficiently long waves. A finite sheet restricts allowed wave numbers, while a supporting foundation adds stiffness and can select a wrinkle wavelength. This is generic continuum mechanics, independent of biological membrane structure.