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Understanding Linear Wave Theory and Movement in Various Water Depths

Explore the mathematical relationships in describing wave movement in deep, intermediate, and shallow water through the Airy wave theory. Learn the expressions for water particle movement under passing waves, essential for sediment transport and coastal geomorphology considerations. Dive into deriving and solving dispersion equations and celerity relationships in different water depths.

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Understanding Linear Wave Theory and Movement in Various Water Depths

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  1. Grades for Assignment 1

  2. Linear (Airy) Wave Theory Mathematical relationships to describe wave movement in deep, intermediate, and shallow (?) water We’ll obtain expressions for the movement of water particles under passing waves - important to considerations of sediment transport --> coastal geomorphology. Works v. well, but only applicable when L >> H Originates from Navier Stokes --> Euler Equations Solution is eta relationship - write eqn. and draw on blackboard - show dependence on x,t Wave Number: k = 2/L Radian Frequency:  = 2/T

  3. Water Surface Displacement Equation What is the wave height? What is the wave period?

  4. Dispersion Equation Fundamental relationship in Airy Theory - put eqns. 5-8, 5-9 on blackboard These are tough to solve, as L is on both sides of equality and contained within hyperbolic trigonometric function. Compilation of Airy Equations - Table 5-2, p. 163 in Komar Door Number 1 = Relationship for wavelength Door Number 2 = Relationship for celerity

  5. Effect of the Hyperbolic Trig Functions on Wave Celerity What’s the relationship for celerity in deep water? What’s the relationship for celerity in shallow water?

  6. So the celerity illustrated is… General Expression: SWS, only depth dependent DWS, T=16 s Gen’l Soln., T=16 s DWS, T=14 s Gen’l Soln., T=14 s Deep-water expression: DWS, T=12 s Gen’l Soln., T=12 s DWS, T=10 s Gen’l Soln., T=10 s Gen’l Soln., T=8 s DWS, T=8 s Shallow-water expression:

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