AMT C24/10

Considerations governing the choice of parameter

The assumptions to be made concerning the development of the pattern are

  1. That the pattern is described by a function $U,V$ of position on the cylinder and of time, satisfying the partial differential equation
    \frac{\partial U}{\partial t}= \phi(\nabla^2) U+I(x,t)U+GU^2-HUV
    V = \psi(\nabla^2) U^2
  2. The operator $\phi(\nabla^2)$ is supposed to take the form $\phi(\nabla^2)=I_2 (1+\nabla^2/k_0^2)^2
  3. The operator $\psi(\nabla^2)$ is supposed to take the form $\psi(\nabla^2)=1/(1-\nabla^2/R_2)$, though in the computations other forms may be used, taking the value zero outside a finite region.
  4. A quasi steady state is assumed to hold, i.e. the time derivative $\partial U/\partial t$ is supposed to be zero, or as so near zero is consistent with slow changes in the radius of the cylinder. This assumption of course implies that certain details as to the effect of the growth on the equation need not be considered.
  5. The function $I(x,t)$ is supposed given in advance. At each time it may be supposed to take the form $I_0-I_2 z^2/l^2$. The quantity $I_0$ is initially supposed to be negative, and to increase to an asymptotic value, reaching very near to it when the optimum wavelength is about one third of a circumference. The quantity $l$ can remain very nearly constant or increase slightly with increasing radius.