Surface WavesSurface TensionDynamic BoundaryBernoulli’s EquationVertical MotionSurface WaveDeep WaterPrins Experiment (1958)Surface WavesSurface WavesSurface TensionSurface TensionWater surface is not fixed.Water surface is not fixed.•Behaves elasticallyBehaves elastically•Based on surface tensionBased on surface tensionSurface tension can be Surface tension can be measured in a thin cylinder.measured in a thin cylinder.•Acts at boundaryActs at boundary•Balanced by gravityBalanced by gravityrghr2cos22for a cylinder with walls at radius r:Dynamic BoundaryDynamic Boundary xppFTBPWater surface is not fixed.Water surface is not fixed.Use a dynamic boundary Use a dynamic boundary condition. condition. •Pressure balanced by Pressure balanced by tensiontension x x+x(x)   )(sin)(sin xxTxTFT   dxdzxx  )(tan)(sinzxdxzdTdxdzTdxdzTFxxxT22 22dxzdTppTBCan be extended to a 2-D surfaceBernoulli’s EquationBernoulli’s EquationMake assumptions about Make assumptions about flow to approximate fluid flow to approximate fluid motion.motion.•IncompressibleIncompressible•InviscidInviscid•IrrotationalIrrotational•Force from gravityForce from gravityApply to Navier-StokesApply to Navier-StokesThe result is Bernoulli’s The result is Bernoulli’s equation.equation.0 v 0P 1fvvtvv0tttvp P1022pgzvt)(22tcpgzvtVertical MotionVertical MotionConsider surface motion.Consider surface motion.•Constant pressureConstant pressure•Velocity relatively smallVelocity relatively small•Vertical velocity Vertical velocity •Vertical deflection hVertical deflection hThe homogeneous equation The homogeneous equation has solutions.has solutions.•Two constants Two constants BB, , CCzzztdtdzwsurfacesurface)(tctg)(22tcttzg022tzgat z = BgBtieCet ),(Surface WaveSurface WaveA sinusoidal surface wave is A sinusoidal surface wave is used to get the speed.used to get the speed.•Separable velocity potentialSeparable velocity potential•Continuity implies Laplace’s Continuity implies Laplace’s equationequation•Find constants of integrationFind constants of integrationA kctkxAtx  sin),(x kctkxzZtzx  cos)(),,(0222 ZkdzZd   kctkxkzkhCtzx  coscosh),,(h khkCgCcktanh22 khkgc tanhkL /2Deep WaterDeep WaterProblemProblemFind the wavelength Find the wavelength LL and and speed speed cc of a wave in deep of a wave in deep water with a period of 3 s.water with a period of 3 s.Begin with a deep water Begin with a deep water approximation, approximation, hh >> >> LL..The speed, period and The speed, period and wavelength are all related.wavelength are all related.•L = cTL = cTSpeed Speed c c = 4.7 m/s= 4.7 m/sWavelength Wavelength L L = 14.1 m= 14.1 m1)/2tanh()tanh(  Lhkh 2tanhgLkhkgc 2gcTc 2gTc Prins Experiment (1958)Prins Experiment (1958)Complicated fluid motion Complicated fluid motion requires experimental requires experimental verification.verification.•Release a bump of water at Release a bump of water at tt = 0 = 0•Sloped shore stops Sloped shore stops reflectionsreflectionsCompare the expected Compare the expected period to experiment.period to experiment.•Discrepancy due to finite Discrepancy due to finite