THE CANONICAL METHOD

The method we have used to solve the simple spring problem is quite general and applies equally well to other layered-medium problems, for example the calculus of Schoenberg-Muir and the Dix equations. For this reason it is worthwhile to pause here and enumerate the steps we used explicitly:

• [1)] Find the general equation describing layer behavior (e.g., equation (5)).
• [2)] Divide the variables that occur in the general layer equation into three classes:
  • [2a)] The layer parameters are assumed given (e.g., the k_i).
  • [2b)] The constant parameters have the same value for all layers in the stack (e.g., F).
  • [2c)] The additive parameters sum through the stack (e.g., the $\Delta x_i$).
• [3)] Rewrite the equation from step 1 so it has the form

  $${\mbox{\rm\bf additive}\,}_i = \mbox{\rm Function}({\mbox{\rm\bf layer}\,}_i) \cdot \mbox{\rm\bf constant}$$ (12)

  (e.g., equation (8)).
• [4)] Identify the term ``$\mbox{\rm Function}({\mbox{\rm\bf layer}\,}_i)$'' as an alternative layer-group representation of the layer parameters (e.g., the 1/k_i). To find the homogeneous equivalent of a stack, convert from the standard representation to the layer group, sum, and convert back.