Kelvin Wedge

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 show that the order- \(n\) Hankel transform in \(r\) diagonalizes the Bessel-type radial operator \(\mathscr{L}_n\), mapping the PDE…

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 consider an axisymmetric pressure distribution \(p(r, \theta)=\bar{p} H(r-l)\) (radial dependence only) and show that its angular Fourier…

The math is sound. The method is not. We consider the radial differential operator for a fixed angular mode \(n\), ∑ Formula

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.

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…

We summarize the ray-theoretic description of Kelvin ship waves in deep and finite depth, deriving the relationship between wavevector and wake angle, the Kelvin wedge, and the…

We derive the ship-wave crest relation \(p=a \cos ^2 \theta\) in deep water from the Doppler-shifted dispersion relation and stationary-phase arguments, interpret \(a\) as an inte…

The article develops a mathematical model for one-dimensional gravity-capillary ship waves, using residue theory to analyze wave components and show how surface tension and wave…

In dispersive water waves (e.g., surface gravity waves on deep or shallow water), several concepts describe how waves travel, how energy moves, and how the wave patterns evolve.…

We summarise Lamb’s treatment of a pressure disturbance on a free surface in a uniform stream, emphasising the contour‐integral evaluation of the far-field wavetrain, the…

We collect the ray-theoretic relations for wave patterns generated by a point source, derive the slope of characteristics and their connection to the group-velocity symbol…

We show how the zeroth-order Hankel transform diagonalises the radial bi-Laplacian, clarify the sign convention \(\nabla^4 \leftrightarrow k^4\), and then apply the same transform…

We recall the continuity and Euler momentum equations in Eulerian form, derive the material derivative ∑ Formula, and explain how Eulerian field descriptions relate to…

This note primarily delves into incompressible flow momentum equation and Surface of water waves and boundary condition and Variational Formulation.