In a Green's function \(G(x,x')\), the primed variable \(x'\) marks the source point — the location of a unit impulse — while \(x\) is the field point where the…

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…

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…

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…

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…