Fluid Dynamics | Part IB, 2004

A layer of water of depth hh flows along a wide channel with uniform velocity (U,0)(U, 0), in Cartesian coordinates (x,y)(x, y), with xx measured downstream. The bottom of the channel is at y=hy=-h, and the free surface of the water is at y=0y=0. Waves are generated on the free surface so that it has the new position y=η(x,t)=aei(ωtkx)y=\eta(x, t)=a e^{i(\omega t-k x)}.

Write down the equation and the full nonlinear boundary conditions for the velocity potential ϕ\phi (for the perturbation velocity) and the motion of the free surface.

By linearizing these equations about the state of uniform flow, show that

where gg is the acceleration due to gravity.

Hence, determine the dispersion relation for small-amplitude surface waves

(ωkU)2=gktanhkh.(\omega-k U)^{2}=g k \tanh k h .

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