Kelvin Wedge Notes

In a Green's function \(G(x,x')\), the primed variable \(x'\) marks the source point — the location of a unit impulse — while \(x\) is the field point where the response is measured. This can be seen as a matrix element \(\langle x | L^{-1} | x' \rangle\) of the inverse operator. For the 1D Poisson equation \(-\frac{d^2 u}{dx^2}=f\), we obtain \(G(x,x')=-\frac12|x-x'|\) using \(\frac{d^2}{dx^2}|x-x'|=2\delta(x-x')\). Additional examples illustrate the same concept.

We show how the spectral relation \(\partial_z \widehat{\varphi}(k, 0)=k \tanh (k h) \widehat{\varphi}(k, 0)\) implies the physical-space operator identity \(\varphi_z(\cdot, \cdot, 0)=(|D| \tanh (h|D|)) \varphi(\cdot, \cdot, 0)\), by inverse Fourier transform and the commutation of \(\partial_z\) with the horizontal Fourier transform.

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 ...

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

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.

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 machinery to an axisymmetric linear free-surface problem to obtain explicit representations for \(\eta(r,t)\) and \(\hat w(s,z,t)\).

We recall the continuity and Euler momentum equations in Eulerian form, derive the material derivative \(\mathrm{d} \mathbf{v} / \mathrm{d} t=\partial_t \mathbf{v}+(\mathbf{v} \cdot \nabla) \mathbf{v}\), and explain how Eulerian field descriptions relate to Lagrangian particlebased descriptions via \(\rho(\mathbf{x}(t), t)\).

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