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 collect closed-form formulas for one- and \(n\)-dimensional Gaussian integrals with linear and oscillatory terms, derive e...

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