non-dimensionalization we will perform the non-dimensionalization of the parameters

prerequisite conditions put velocity potential function as follow \[ \begin{equation} \phi=U x+\varphi-\frac{1}{2} U^2 tz \en...

In dispersive water waves (e.g., surface gravity waves on deep or shallow water), several concepts describe how waves travel,...

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： 1. 表示直接推导或结果 连接词/短语： Therefore Example: "Therefore, Equation (3) can b...

paper core

Pressure to the surface to a stream notation \(\varpi\): -> r ///////p439 (22) \(\sigma\): -> \(\omega\) \(c\): -> \...

wave patterns 📄P411, Sec 12.4 For a point source \(P\), the characteristics carrying disturbances pass through \(P\) and we...

1-dimensional Gaussian integral \[ \begin{gathered} \int_{-\infty}^{\infty} e^{-x^2} d x=\sqrt{\pi} \\ \int_{-\infty}^{\infty...

Solution Explanation show that, under the (zeroth-order) Hankel transform in the radial variable, \[ \mathcal{H}_0\left[\nabl...

This part mainly focus on ship wave with the infinite depth water, also mainly use Fourier transform method.