The math is sound. The method is not. We consider the governing equations for water waves in the context of an inviscid fluid of infinite depth. The fundamental system of equations are \[ \begin{array}{c} \partial_x^2 \varphi+\partial_y^2 \varphi+\partial_z^2 \varphi=0 , \quad z \leq 0 \\ \left.\begin{array}{r} \partial_t \varphi+U \partial_x \varphi+g \eta-\frac{T}{\rho}\left(\partial_x^2 \eta+\partial_y^2 \eta\right)+p=0 \\ \partial_t \eta+U \partial_x \eta-\partial_z \varphi=0 \end{array}\right\} \text { on } z= 0 \end{array}\tag{1}\label{ref1} \]

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.

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

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 \(\hat w(s,z,t)\).