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worksheet10 - Fluid Dynamics 3 2011/12 Sheet 10 Homework to...

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Fluid Dynamics 3 2011/12 Sheet 10 Homework to be handed in Friday 20th January: questions 2,5. 1. Consider small amplitude two-dimensional oscillations on the free-surface of an unbounded fluid of infinite depth. At large depths, the fluid is a rest. The elevation of the free-surface is given by z = ζ ( x, t ), and the fluid occupies z < ζ . (i) Assuming incompressibility and an irrotational flow, write down the linearised equations of motion in terms of ζ and a velocity potential φ ( x, z, t ). (ii) Assume a time-harmonic dependence with angular frequency ω and wavenumber k of the form ζ ( x, t ) = H sin( kx - ωt ) and hence show that the corresponding velocity potential is φ ( x, z, t ) = - ω k H e kz cos( kx - ωt ) In doing so, find the dispersion relation for deep water, ω 2 = gk. (iii) Assume in that H 1, so that small amplitude waves are supposed to travel in the positive x -direction. Now consider the paths of fluid particles which are initially located at x = ( x 0 , z 0 ). Denoting their subsequent position by x = ( x 0 + X ( t ) , z 0 + Z ( t )) and on the assumption that the displacements from the initial position are small (i.e | X/x 0 | 1, | Z/z 0 | 1), show that the paths taken approximately satisfy X 2 ( t ) + Z 2 ( t ) = H 2 e
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