Fluid Mechanics

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…

We derive the ship-wave crest relation \(p=a \cos ^2 \theta\) in deep water from the Doppler-shifted dispersion relation and stationary-phase arguments, interpret \(a\) as an inte…

The article develops a mathematical model for one-dimensional gravity-capillary ship waves, using residue theory to analyze wave components and show how surface tension and wave…

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.…

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…

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…

We show how the zeroth-order Hankel transform diagonalises the radial bi-Laplacian, clarify the sign convention \(\nabla^4 \leftrightarrow k^4\), and then apply the same transform…

We review the Eulerian continuity and Euler momentum equations, clarify the relationship between Eulerian and Lagrangian descriptions via the material derivative, and briefly…

We recall the continuity and Euler momentum equations in Eulerian form, derive the material derivative ∑ Formula, and explain how Eulerian field descriptions relate to…

This note primarily delves into incompressible flow momentum equation and Surface of water waves and boundary condition and Variational Formulation.