Rectangular Coordinates, Strip and Semi-strip, steady.

by K. D. Cole and D. H. Y. Yen

The GF for the 2D strip satisfies:

 + = - (x - x)(y - y) (18)

For the infinite strip, shown in Fig. 1a, the domain is (- < x < ,  0 < y < W) and the boundary conditions are

 ki + hiG = 0 at y = 0 and y = W G,   are bounded as x (19)

For the semi-infinite strip, shown in Fig. 1b, the domain is (0 < x < ,  0 < y < W)and the homogeneous boundary conditions are
 ki + hiG = 0 at x = 0,  y = 0, and y = W G,   are bounded as x + (20)

The GF of infinite and semi-infinite strip are described by number XI0YKL which represents 36 different GF for I = 0, 1, 2, or 3 and K, L = 1, 2, or 3. Note that the GF for I = 0 are for the infinite strip geometry only.

The GF for the strip has a single-summation form with eigenfunction Yn, eigenvalue , norm Ny, and kernel function Pn, as follows:

 G(x, y  x, y) = P0(x, x) + Pn(x, x) (21)

The first term with kernel function P0 is needed only when Y22 is part of the GF number (i.e., when zero is an eigenvalue). There are nine different eigenfunctions associated with the nine possible boundary condition combinations YKL (K, L = 1, 2, or 3). Table 1 contains the eigenfunctions and norms, and Table 2 contains the associated eigenconditions (and eigenvalues for simple cases).
Kernel functions Pn for n 0 are given by

 Pn(x, x) = S+exp(- x - x) + S-exp(- x + x) (22)

where the values for S+ and S- are given in Table 3. Kernel functions Pn for n 0 have four different forms, one for each of the boundary condition combinations XI0 for I = 0, 1, 2, and 3.

Kernel functions P0 satisfy

 = - (x - x) (23)

as well as appropriate homogeneous boundary conditions. Functions P0, listed in Table 4, must be included in the GF whenever zero is an eigenvalue (XI0Y22 for I = 0, 1, 2, and 3).

Table 1. Eigenfunctions and inverse norm a, b
 Geometry Yn(y) Ny-1 Y11 sin(y) 2/W Y12 sin(y) 2/W Y13 sin(y) 2/W Y21 cos(y) 2/W Y22 cos(y); 0 2/W for  0 1; = 0 1/W for  = 0 Y23 cos(y) 2/W Y31 Wcos(y) + (h1W/k)sin(y) 2/W Y32 Wcos(y) + (h1W/k)sin(y) 2/W Y33 Wcos(y) + (h1W/k)sin(y) 2/W

a Index n = 1, 2,... for all cases except Y22 with n = 0, 1, 2,...
b = [(W)2 + (hiW/k)2] ÷ [(W)2 + (hiW/k)2 + hiW/k]
= ÷ [(W)2 + (hiW/k)2 + (h1W/k)]

Table 2. Eigencondition and eigenvalues for Yn(y)a
 Geometry Eigencondition Eigenvalues Y11 sin(W) = 0 n/W Y12 cos(W) = 0 (2n - 1)/2W Y13 Wcot(W) = - h2W/k Y21 cos(W) = 0 (2n - 1)/2W Y22 sin(W) = 0 n/W, n = 0, 1, 2,... Y23 Wtan(W) = h2W/k Y31 Wcot(W) = - h1W/k Y32 Wtan(W) = hW/k Y33 tan(W) = [(h1 + h2)/k]/[ - h1h2k-2]

a Index n = 1, 2,... for all cases except Y22 with n = 0, 1, 2,...

Table 3. Coefficients for Pn for n 0, strip and semi-strip.
 Geometry S+ S- X00 1 0 X10 1 -1 X20 1 1 X30 W + hW/k W - hW/k

Table 4. Kernel function P0 for strip and semi-strip.
 Geometry P0(x, x) X00 - x - x X10 - x - x + x + x X20 - x - x - x + x X30 - x - x + x + x + k/h