1. Surface Waves

2. 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 :

3. 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

4. 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.

5. 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 = 

6. 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

7. 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

8. 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