1-3 Appendix B

For $k_{⊥} = k_{⊥} (ω, X) = k \sin ψ$ and $k_{∥} = k_{∥} (ω, X) = k \cos ψ$ , the Jacobian is given by

\begin{matrix} (B1) & | J (\frac{k_{⊥}, k_{∥}}{ω, X}) | = | \begin{matrix} \frac{\partial k_{⊥}}{\partial ω} & \frac{\partial k_{⊥}}{\partial X} \\ \frac{\partial k_{∥}}{\partial ω} & \frac{\partial k_{∥}}{\partial X} \end{matrix} | \end{matrix}

Using the standard rules for manipulating derivatives (e.x., Chain rule) gives

\begin{matrix} (B2) & {\frac{\partial k_{⊥}}{\partial ω} |}_{X} = \sin ψ {\frac{\partial k}{\partial ω} |}_{X} \end{matrix}

and

\begin{array}{r} (B3) & {\frac{\partial k_{∥}}{\partial ω} |}_{X} = \cos ψ {\frac{\partial k}{\partial ω} |}_{X} + k {\frac{\partial (\cos ψ)}{\partial X} |}_{ω} \\ (B4) & = \cos ψ {\frac{\partial k}{\partial ω} |}_{X} - k \sin ψ \cos^{2} ψ \end{array}

Similarly,

\begin{aligned} (B5) & {\frac{\partial k_{⊥}}{\partial X} |}_{ω} & = \sin ψ {\frac{\partial k}{\partial X} |}_{ω} + k \cos^{3} ψ \\ (B6) & {\frac{\partial k_{⊥}}{\partial X} |}_{ω} & = \cos ψ {\frac{\partial k}{\partial X} |}_{ω} \end{aligned}

Combining these results gives the Jacobian as

\begin{matrix} (B7) & | J (\frac{k_{⊥}, k_{∥}}{ω, X}) | = - k \cos^{2} ψ {\frac{\partial k}{\partial ω} |}_{X} \end{matrix}

The derivative term, $\frac{\partial k}{\partial ω}$ , in the above equation can be found by differentiating the dispersion relation $D (k, ω, X) = 0$ to give

\begin{matrix} (B8) & \frac{d D}{d ω} = \frac{\partial D}{\partial ω} + \frac{\partial D}{\partial k} \frac{\partial k}{\partial ω} = 0 \end{matrix}

\begin{matrix} (B9) & | J (\frac{k_{⊥}, k_{∥}}{ω, X}) | = k \cos^{2} ψ \frac{\partial D}{\partial ω} {(\frac{\partial D}{\partial k})}^{- 1} \end{matrix}

Thus in 1-3 Appendix A, one find

\begin{aligned} N (ω) & = \frac{1}{2 π^{2}} \int_{X_{min}}^{X_{max}} g (X) | J (\frac{k_{⊥}, k_{∥}}{ω, X}) | k_{⊥} d X \\ (add) & = \frac{1}{2 π^{2}} \int_{X_{min}}^{X_{max}} \frac{g (X) k^{2}}{(1 + X^{2})^{\frac{3}{2}}} \frac{\partial D}{\partial ω} {[\frac{\partial D}{\partial k}]}^{- 1} X d X \end{aligned}