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

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.

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回乡偶书二首贺知章少小离家老大回，乡音无改鬓毛衰。儿童相见不相识，笑问客从何处来。离别家乡岁月多，近来人事半销磨。唯有门前镜湖水，春风不改旧时波。清心瘦清泉