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…

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…

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…

claim: ∑ Formula

We consider the governing equations for water waves in the context of an inviscid fluid of infinite depth. The fundamental system of equations are ∑ Formula

We introduce azimuthal Fourier expansion for functions with circular symmetry: we write \(f(r, \phi)\) as a sum of angular modes ∑ Formula, where ∑ Formula is obtained by integrati…

Hankel–Bessel Ladder Identities Claim. For suitable \(f_m(r)\) and \(k>0\), ∑ Formula Here ∑ Formula.

This note introduces Ninja, PyTorch’s MPS backend, and NVIDIA DALI as tools relevant to setting up a PINN workflow. It outlines their roles and the hardware constraints to…

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…