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Showing posts with the label Dispersion Relation

Coastal Engineering: Linear Wave Theory and Dispersion Mechanics

 Small-amplitude water wave kinematics are modeled using Airy Linear Wave Theory. The surface elevation profile $(\eta)$ of a progressive wave traveling in the x-direction is given by: $\eta = a \cdot \cos(k \cdot x - \omega \cdot t)$ ​Where $a$ is wave amplitude, $k$ is wave number $(k = 2\pi / L),$ and $\omega$ is angular frequency $(\omega = 2\pi / T).$ The relationship between wave frequency, water depth (d), and wavelength (L) is governed by the Linear Wave Dispersion Relation: $\omega^2 = g \cdot k \cdot \tanh(k \cdot d)$ ​In deep water $(d/L > 0.5),$ $\tanh(k \cdot d) \to 1,$ reducing wave celerity to $C_0 = \frac{g \cdot T}{2\pi}.$ In shallow water $(d/L < 0.05)$, wave speed depends solely on water depth: $C = \sqrt{g \cdot d}.$ ​Monsoonal storm surges and cyclonic wave actions along India's coastline (e.g., eastern Bay of Bengal) cause severe coastal erosion and port infrastructure damage. ​Modern coastal engineering projects integrate wave transformation models (suc...