Recent notes

Water waves, mathematics & computation

Bing-Hai Lei’s notes on theoretical fluid mechanics, numerical methods, and the mathematics behind them—derivations, reading trails, and reusable calculations.

Kelvin Wedge Reading NotesWhat does $x^{\prime}$ mean in a Green’s function? — a linear algebra and operator view

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…

2026-04-30

人应该怎样度过一生一

浓雾氤氲的森林，分不清是清晨还是黄昏，天边应该是有丝丝破碎的晨曦亦或是夕阳，还未走进你眼眸，便融化进了昏霭，染成一片模糊的光晕。你的思绪也仿若虚无之地，白茫茫一片，犹豫，抬起脚，犹豫，犹豫，犹豫，向前？向后？是追忆还是邂逅迷茫？

2026-03-25

Kelvin Wedge Reading NotesFrom the Spectral Dirichlet-Neumann Symbol to the Physical-Space Operator Identity

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…

2026-02-05

Kelvin Wedge Negative ResultsHankel Diagonalization of the Radial Bessel Operator

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…

2026-01-29

Kelvin Wedge Negative ResultsFourier Coefficients of Quadratic Forms in $\cos \gamma$ for Steady Capillary-Gravity Waves

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.…