One-dimensional sinusoidal surface

In this section, situations are considered for which the surface of a stressed solid is initially flat, or nearly so, and for which the slope of the evolving surface is everywhere small in magnitude throughout the evolution process. Chemical potential for a one-dimensional sinusoidal surface shape was developed in Section 8.4.1, for a two-dimensional sinusoidal shape in Section 8.5.3, and for a general small amplitude surface profile in Section 8.5.2. These results are used to examine surface evolution by either the mechanism of surface diffusion or condensation, as described in Section 9.1. In all cases considered in this section, surface energy is assumed to have the constant value 70, independent of surface orientation and surface strain. Implications of surface energy anisotropy and strain dependence are examined subsequently. 

Consider a biaxially stressed solid with a free surface described in a rectangular coordinate system by its height 

Assuming that the sinusoidal surface evolves into a sinusoidal surface of the same wavelength under generalized plane strain conditions, the normal speed of the evolving surface with respect to the material instantaneously 

The speed is positive locally if material is being added there. Substitution of the normal surface speed Vn and chemical potential x for this particular case into the general relationship (9.9) governing surface diffusion 5helds the ordinary differential equations 

The sinusoidal surface is stable or unstable, depending on whether a grows indefinitely with time or decays continuously with time. The value of the wavelength A discriminating between these two possibilities is that value 

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