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Surface Waves
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Surface Tension Water surface is not fixed. Behaves elastically Based on surface tension Surface tension can be measured in a thin cylinder. Acts at boundary Balanced by gravity for a cylinder with walls at radius r :
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Dynamic Boundary Water surface is not fixed. Use a dynamic boundary condition. Pressure balanced by tensionPressure balanced by tension x x+ x (x)(x) z Can be extended to a 2-D surface
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Bernoulli’s Equation Make assumptions about flow to approximate fluid motion. Incompressible Inviscid Irrotational Force from gravity Apply to Navier-Stokes The result is Bernoulli’s equation.
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Vertical Motion Consider surface motion. Constant pressureConstant pressure Velocity relatively smallVelocity relatively small Vertical velocity Vertical velocity Vertical deflection hVertical deflection h The homogeneous equation has solutions. Two constants B, CTwo constants B, C at z =
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Surface Wave A sinusoidal surface wave is used to get the speed. Separable velocity potentialSeparable velocity potential Continuity implies Laplace’s equationContinuity implies Laplace’s equation Find constants of integrationFind constants of integration A x h
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Deep Water Problem Find the wavelength L and speed c of a wave in deep water with a period of 3 s. Begin with a deep water approximation, h >> L. The speed, period and wavelength are all related. L = cT Speed c = 4.7 m/s Wavelength L = 14.1 m
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Prins Experiment (1958) Complicated fluid motion requires experimental verification. Release a bump of water at t = 0 Sloped shore stops reflections Compare the expected period to experiment. Discrepancy due to finite depth R Q H next
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