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: \[ \begin{aligned} & {\widehat{\left(\partial_x f\right)_n}}(k)=\frac{k}{2}\left(\hat{f}_{n-1}(k)-\hat{f}_{n+1}(k)\right) \\ & {\widehat{\left(\partial_y f\right)_n}}(k)=\frac{k}{2 i}\left(\hat{f}_{n-1}(k)+\hat{f}_{n+1}(k)\right) \end{aligned}\tag{1} \]

The math is sound. The method is not. We consider the governing equations for water waves in the context of an inviscid fluid of infinite depth. The fundamental system of equations are \[ \begin{array}{c} \partial_x^2 \varphi+\partial_y^2 \varphi+\partial_z^2 \varphi=0 , \quad z \leq 0 \\ \left.\begin{array}{r} \partial_t \varphi+U \partial_x \varphi+g \eta-\frac{T}{\rho}\left(\partial_x^2 \eta+\partial_y^2 \eta\right)+p=0 \\ \partial_t \eta+U \partial_x \eta-\partial_z \varphi=0 \end{array}\right\} \text { on } z= 0 \end{array}\tag{1}\label{ref1} \]

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.

Knowledge related to the UK driving test