We introduce azimuthal Fourier expansion for functions with circular symmetry: we write \(f(r, \phi)\) as a sum of angular modes \(\hat{f}_n(r) e^{i n \phi}\), where \(\hat{f}_n(r)\) is obtained by integrating over \(\phi\) and measures the strength of mode \(n\) at radius \(r\). We then contrast this discrete angular Fourier series on \([0,2 \pi]\) with the usual continuous Fourier transform on \((-\infty, \infty)\), and we briefly note that this separation of radial and angular dependence is especially useful in optics, quantum mechanics, fluid dynamics, and image processing.

Hankel–Bessel Ladder Identities Claim. For suitable \(f_m(r)\) and \(k>0\), \[ \mathscr{H}_{m+1}\!\left[\left(\partial_r-\frac{m}{r}\right) f_m\right](k) =-\,k\,\hat{f}_m(k). \tag{1} \] Here \(\hat f_m(k):=\mathscr H_m[f_m](k)=\displaystyle\int_0^\infty f_m(r)\,J_m(kr)\,r\,dr\).

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 ...

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.