Figure 4: Undulation instability.
(a) Equilibrium wrinkle patterns in a flat thin ribbon residing on a flat body of water (simulation), held at its far ends with a fixed wavy boundary. The preferred wrinkle wavelength λ0 is 5% shorter than the boundary wavelength λB. Shown are the results for various values of the predicted undulation wavelength 
(equation (19)), where the ribbon’s width is used as a surrogate for the system scale d. (b) The measured undulation wavelength is plotted against the predicted one. While corroborating the predicted scaling, the measured wavelength is consistently larger than 
by a factor of ≈1.7, suggesting that the relevant system scale d may be different than the ribbon width by a constant factor. (c) Simulation results for a spherical cap with free boundary conditions. An undulation appears spontaneously, and its observed wavelength (black triangles) agrees with the predicted value, where the shell’s diameter appears to be a good surrogate for the system scale d.
