c10 pre school assignment 2-q1 nov 2011_f
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Question1Derive the solution of 2D monochromatic waves of finite water depth i.e. the solution to
∆ 0
With
, ,
Since two‐dimensional analysis results in partial derivatives with respect to z set to zero, then the
equation of continuity for incompressible flow, may be expressed using wave velocity components
u w0Substitutingforu& →
Resulting in the LaPlace equation for velocity potential
0
It can be seen that this analysis satisfies the condition of irrotationality applied in two dimensions
0 →
0 … 1
Including the velocity potential term when applying the principle of conservation of momentum, and
neglecting (due to small velocities) terms representing non‐linear kinetic energy, gives (from
Bernoulli)
gy
0 … 2
Equations 1 and 2 may then be solved through the application of appropriate boundary conditions
equations where, in the y direction, the ‘free surface velocity’ may be equated to the velocity of the
fluid, such that
u… 3
Where is the amplitude of the wave at any location throughout , and is a function of x and t (time). The assumption of low amplitude waves simplifies application of this boundary condition,
since the non‐linear term is small and may be neglected. Further this assumption allows application
at y=0: hence,
0 … 4
Equation 2 is also simplified, since at the free surface atmospheric pressure equates p to zero, and
wave amplitude is : hence
gy
0 0 … 5
Finally at the seabed, the vertical velocity component, w, may clearly be set to zero giving
0 … 6
From equation 1, the progressive wave expression for velocity potential may be presented as
Φ cos … 7
Where W,k,h and are constants. Through first and second‐order partial differential of equation 7, coupled with appropriate application of boundary conditions as defined in equations 3 to 7 for a
periodic wave form , it will be seen that
, ;
sinh sinh
Where
tanh tanh
And
2
Where k is known as the wave number.
Substitutions for W, k and h into equation 7 thus yield the equation for velocity potential.
Φ
coshcos