Fluid Mechanics

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 response is…

We show how the spectral relation ∑ Formula implies the physical-space operator identity ∑ Formula, by inverse Fourier transform and the commutation of \(\partial_z\) with the hori…

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 ∑ Formula.…

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: ∑ Formula

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…

The math is sound. The method is not. We introduce azimuthal Fourier expansion for functions with circular symmetry: we write \(f(r, \phi)\) as a sum of angular modes ∑ Formula, whe…

The math is sound. The method is not. Hankel–Bessel Ladder Identities Claim. For suitable \(f_m(r)\) and \(k>0\), ∑ Formula Here ∑ Formula.

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We study the free-surface response to a moving pressure point over finite depth using 2D Fourier transforms, contour deformation, and a radiation condition. By analysing poles…