Example Doubly periodic surface perturbation

The deformation fields arising from any small amplitude surface perturbation can be constructed from fields corresponding to doubly periodic perturbations by Fourier methods. Examine the particular case of a doubly periodic perturbation described by 

Determine the necessary condition on the wavelengths for the surface to be stable for the perturbation (8.93). 

The change in elastic energy of the system per period due to formation of the perturbed shape is given by the surface integral representing the work done in 

Similarly, the change in surface energy due to formation of the perturbed shape is 

Several observations can be made on the basis of these simple three-dimensional results. 

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