Boundary Value Problems in Two Dimensions

Consider the following problem: A vibrating system has an amplitude response $\psi$ to a source function $f$ which is governed by the inhomogeneous Helmholtz equation

$$(\nabla^2+k^2)\psi(\vec x)=-f(\vec x)~.$$ (559)

Assume that this equation applies to a 2-dimensional region $R$ whose boundary is designated by $\partial R$ . Suppose that on this boundary the response amplitude satisfies the inhomogeneous mixed Dirichlet-Neumann boundary condition

$$\left[ a(\vec x)\psi(\vec x)+\vec n\cdot \vec\nabla \psi (\vec x) \right]_{\partial R}=g(\vec x)\vert _{\partial R}~.$$ (560)

Find the response amplitude $\psi(\vec x)$ !

This problem is characterized by

1. the shape of the as-yet-unspecified region $R$ ,
2. the as-yet-unspecified inhomogeneities $f$ and $g$ , and
3. the as-yet-unspecified effective stiffness of the boundary, the function $a(\vec x)$ .

Thus, by omitting reference to the particular measurement of these properties, one has mentally subsumed a vast number of particular problems, which govern the response of a vast number of linear systems, into a new concept ⁵¹, an equivalence class of problems. A representative class member is characterized by Eqs.(5.59) and (5.60).

Footnotes

... concept⁵¹

It is worthwhile to point out that the process of measurement omission is the process by which all concepts are formed. This observation and the procedure for implementing this process were first spelled out by Ayn Rand in Chapters 1-2 of Introduction to Objectivist Epistemology, 2nd Edition, edited by H. Binswanger and L. Peikoff. Penguin Books, Inc., New York, 1990.