We show how the spectral relation \(\partial_z \widehat{\varphi}(k, 0)=k \tanh (k h) \widehat{\varphi}(k, 0)\) implies the physical-space operator identity \(\varphi_z(\cdot, \cdot, 0)=(|D| \tanh (h|D|)) \varphi(\cdot, \cdot, 0)\), by inverse Fourier transform and the commutation of \(\partial_z\) with the horizontal Fourier transform.

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. We factor the quadratic denominator \(a+b \cos \gamma+c \cos ^2 \gamma\) and reduce the computation of its Fourier coefficients to those of \(1 /(\cos \gamma-\lambda)\). Solving the associated second-order recurrence for the Fourier coefficients yields a closed-form expression in terms of \(\lambda_{ \pm}\)and \(q_{ \pm}=\lambda_{ \pm}+i \sqrt{1-\lambda_{ \pm}^2}\), leading to the final explicit formula for \(c_m(k)\) in the steady ship-wave setting.

The math is sound. The method is not. We recast the azimuthal mode-coupled surface equations as a block-Toeplitz convolution in the mode index and diagonalize this coupling via a discrete Fourier transform in \(n\), then briefly review the basic definition and core properties of the DFT used in this step.

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