Solutions for circular cross-section

The matrices $F$ and $V$ defined in (2.80) can be found analytically for circular cross-section using the standard integral in equation (A.1) of appendix A. A full derivation is given in appendix B. The result is that each element is a function of $\beta = R_1/R_2$ and the element $(n,m)$ is given by

$$F_{nm}(\beta) = \frac{2 \beta \gamma_m J_1(\beta \gamma_m)} {(\beta^2 \gamma_m^2 - \gamma_n^2)J_0(\gamma_m)}$$ (2.85)

where $\beta = R_1/R_2$ and $F(0,0)=1$ and

$$V_{nm}(\beta) = F_{nm}(1/\beta)$$ (2.86)

When the change in cross-section tends to zero (ie. $\beta$ tends to 1) we obtain

$$F \approx I - \epsilon Q$$ (2.87)

with $I$ being the identity matrix (a diagonal matrix with all the entries having a value of 1). $\epsilon$ is the fractional change in cross-section, $\epsilon = (S_2 - S_1)/S_1$ and $Q$ is a matrix whose elements are given by

$$Q_{nm} = \left\{ \begin{array} {r@{\quad:\quad}l} 0 & n = ... ...m^2 - \gamma_n^2} & \mbox{otherwise}. \\ \end{array} \right.$$ (2.88)