The wave number k of the one-dimensional problem has become the separate
wave numbers kx for f (x) and ky for g(y). We show later how these are related.
(See also Problem 18 at the end of this chapter.)
The continuity condition on ψ(x, y) requires that the solutions inside and
outside match at the boundary. Because ψ = 0 everywhere outside, the continuity
condition then requires that ψ = 0 everywhere on the boundary. That is,