Regular waves

Question 1

Assumptions

Answer 1

1. conservation of mass
2. irrotationality
3. a&laquo_space;d
4. a«L
5. energy conservation

3/4 allow non linear effects to be ignored

Question 2

Governing equation

Answer 2

Laplace because of the the conservation of mass and no vorticity

partial d2u/dx2 + partial d2u/dz2 = 0

this leads to a solution based on

u =sum to N of ak^n.f_n(z).sin(n(omega.t -k.x))

from the small amplitude assumptions ak «1 and only the linear part of u is adopted

Question 3

Kinematic Free Surface BC

Answer 3

Fluid particle at the surfce should remain there or the surface velocity normal to the surface is equal to the particles velocity normal to the surface

leads to:

partial d.eta/dt.cost(theta) = wcos(theta) - usin(theta)
w = partial d.eta/dt + u(partial d.eta/dx)

For linear theory the first term in w&raquo_space; than the second

Question 4

Dynamic free surface BC

Answer 4

Pressure at the surface is constant and equal tot he atmospheric pressure
-neglects the affect of overlying air flow
-leads tot he unsteady bernoulli equation

P = rho.(partial d phi/dt) - rho.g.z + (u^2 + w^2)/2g + Const.

the first terms are linear with the third being nonlinear

Question 5

Nonlinearality in loading

Answer 5

Both energy and force are nonlinear
- observed surface profiles have higher sharper peaks and broader shallower troughs than expected

Crest-trough Asymmetry

Question 6

Drift velocity

Answer 6

Particle orbit definitions:

u = partial d zeta/dt
w = partial d eta/dt
- elliptical in shallow water and circular in deep water

In 2nd order there is lagrangian drift

(u)_L = u(x+zeta,z+eta,t)
- leads to open particle orbits witt he largest effects near the surface
- net forward drift isnt realistic, there is no net when it is balanced by the uniform return flow
-Induced streaming leads to +ve near bed velocities which quickly becomes -ve with increasing z before becoming positive again

Question 7

motivation for stokes solutions

Answer 7

-Increased steepness makes higher order terms more important with more and more terms needing to be included
*nonlinear regular waves cant capture all nonlinearality
-Stokes is a series solution based on small perturbation expansions

Question 8

2nd order surface profile

Answer 8

Crest-trough asymmetry
-change in mean water level
-change in mean velocity but not the lagrangian component

2nd order stokes incorporates 2nd harmonic which is bound

Question 9

Bound waves

Answer 9

-tied to free waves
-doesnt satisfy dispersion equation
-its fit to the FSBC’s are really bad

Question 10

Numerical regular wave solutions for eta

Answer 10

-series expansion is summed numerically
-appropriate for non breaking waves
- based on the solution of a boundary value problem

Stokes expansion steps
1.apply steady frame of reference by removing t
2.alwasy satisfy grad2 phi = 0 (governing equation) and partial dphi/dy=0 @ y=0 (bottom boundary)
3.unknown are Aij and bij are found by least square fit to the full nonlinear FSBCs

Pattern of harmonics for phase velocity and uniform current
-odd orders of stokes use all the odd ones before it
-even orders use all the even ones before and the mean

Question 11

Numerical regular wave solutions for stream function

Answer 11

-widely used
-assumes a steady wave and is base on a steady frame of reference
-automatically satisfies the governing equation and the bottom BC
-unknowns are now Xn, once again determined by the least squares fit to the full nonlinear FSBC
-Applicable to a wide range of water depths
* predict eta(x) and phi(x,z) and therefore u/w from H,T and d
* predict stream function(x,z), trident, therefore u/w from a measured eta(x)

• based on the solution of a boundary value problem

Question 12

SAWT limitaions

Answer 12

can cover nonlinearity, unsteadiness or directionality