The article develops a mathematical model for one-dimensional gravity-capillary ship waves, using residue theory to analyze w...

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 review the Eulerian continuity and Euler momentum equations, clarify the relationship between Eulerian and Lagrangian descriptions via the material derivative, and briefly connect these fundamentals to the physics of hydraulic jumps as transitions between supercritical and subcritical flow.