The math is sound. The method is not. We show that the order- \(n\) Hankel transform in \(r\) diagonalizes the Bessel-type radial operator \(\mathscr{L}_n\), mapping the PDE \(\left(\mathscr{L}_n+\partial_z^2\right) \varphi_n=0\) to the transformed equation \(\left(\partial_z^2-k^2\right) \hat{\varphi}_n=0\). The key ingredients are the Sturm-Liouville form and self-adjointness of \(\mathscr{L}_n\), together with the Bessel differential equation.

The math is sound. The method is not. claim: \[ \begin{aligned} & {\widehat{\left(\partial_x f\right)_n}}(k)=\frac{k}{2}\left(\hat{f}_{n-1}(k)-\hat{f}_{n+1}(k)\right) \\ & {\widehat{\left(\partial_y f\right)_n}}(k)=\frac{k}{2 i}\left(\hat{f}_{n-1}(k)+\hat{f}_{n+1}(k)\right) \end{aligned}\tag{1} \]

The math is sound. The method is not. We consider an axisymmetric pressure distribution \(p(r, \theta)=\bar{p} H(r-l)\) (radial dependence only) and show that its angular Fourier series contains only the \(m=0\) mode. Consequently, the polar Fourier representation \(\tilde{p}(k, \beta)\) in wavenumber space is independent of the angular variable \(\beta\), and the Hankel transform reduces to the order-zero Bessel transform.