人生无聊，但你要变得有趣！

basic concept ninja ninja is a build system—a tool that actually runs the compile and link commands for a software project. T...

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, branch points, and stationary points of the dispersion curve \(G(\alpha, \beta)=0\), we derive an asymptotic representation of \(\eta\) via residues and stationary-phase contributions and clarify the relevant complex-analytic singularity structure.

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 scaling of amplitude along rays via wave action conservation, and we connect these to Froude numbers and interference effects between bow and stern waves.

We derive the ship-wave crest relation \(p=a \cos ^2 \theta\) in deep water from the Doppler-shifted dispersion relation and ...