在 MathJax 中，aligned、align 和 align* 是用于对齐多行数学公式的环境，但它们在功能、使用场景和编号规则上有明确区别： 功能与使用场景 align 环境 用于独立排版多行对齐的数学公式，每行默认自动编号。适用于需要逐行编号...

在英文数学论文中，推导公式时连接前后式子的常用连接词和短语可以分为以下几类，以确保逻辑清晰、推导严谨：

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. Below is an overview of wave fronts, wave crests, the wave number vector \(\mathbf{k}\), phase velocity \(\mathbf{c}_p\), group velocity \(\mathbf{c}_g\), and rays, along with how their directions compare.

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 underlying simple-harmonic free-surface condition, a Bessel–delta normalisation, and a geometric construction for envelopes of straight lines.

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 \(G\), and clarify the geometric roles of wavefronts, wave vectors, and rays, culminating in a phase-distance relation \(\theta=k \cos \mu r\) for capillarygravity wakes.