c10 pre school assignment 2-q1 nov 2011_f

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Question 1 Derive the solution of 2D monochromatic waves of finite water depth i.e. the solution to 0 With ݐ ,ݕ ,ݔ ݑ ݔ ݓ ݕSince twodimensional 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 ݔ w ݕ0 Substituting for u & →ݓ ݔ ݔ ݕ ݕResulting in the LaPlace equation for velocity potential ݔ ݕ0 It can be seen that this analysis satisfies the condition of irrotationality applied in two dimensions ݓ ݔ ݑ ݕ0 ݔ ݕ ݔ ݕቇൌ0 … ݑݍܧݐ 1 Including the velocity potential term when applying the principle of conservation of momentum, and neglecting (due to small velocities) terms representing nonlinear kinetic energy, gives (from Bernoulli) ߩ ߩ gy ݐ0… ݑݍܧݐ 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 ߟ ݐu ߟ ݔ ݓ ݕݑݍܧ …ݐ 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 nonlinear term is small and may be neglected. Further this assumption allows application at y=0: hence, ߟ ݐ ݕ ݕݐൌ0 … ݑݍܧݐ 4

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Page 1: C10 Pre School Assignment 2-Q1 Nov 2011_F

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 

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 

Page 2: C10 Pre School Assignment 2-Q1 Nov 2011_F

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  

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