An isolated surface mound

Suppose that the initial surface shape is a shallow mound, circular in plan view, that is described by the rotationally symmetric function 

The Fourier transform of the initial shape in normalized variables is 

A comparable coordinate transformation to polar coordinates, say f and 0, in the physical plane is 

The cosine factors in the integrand are then rewritten in terms of polar coordinates by means of the identity 

The only place in the integrand where ip appears is through the factor cos2A (/3 in each term of the infinite series in (9.42). The integral of each such term over ip is zero. Consequently, the solution is independent of 6, as was expected, and it takes the form 

---

Fig. 9.3. The initial cross-sectional profile t = 0.0 of an isolated surface mound, as given by (9.38), and the shapes of the mound resulting from mass transport by surface diffusion at two later times are illustrated for the case when a = Ugr, which exhibits the most rapid growth. The profiles for times t = 0.5 and 1.0 were...

---