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WATER WAVE MECHANICS
- Linear wave theory 41 a. Lateral boundary conditions b. Laplace equation c. Progressive wave d. Dispersion
- Engineering wave properties 78 a. Particle kinematics progressive waves b. Water particle trajectory c. Pressure field under a progressive wave d. Energy in a wave field e. Group velocity Cg f. Wave transformation: refraction and diffraction
- Wave records and wave spectra 187
- Spectral wave modelling a. The WAMC4 Model
- Nonlinear properties derived from linear theory 295 a. Lagrangian mass transport b. Radiation stress
- Nonlinear waves 295
- Phase resolving models CHARACTERISTICS OF WAVES -For describing a wave the important parameters are the length (L), the height (H) and the water depth over which they are propagating (h). Speed of the wave= celerity C=L/T coefficient of surface tension as σ Water surface elevation η
LINEAR WAVE THEORY
Boundary conditions The governing Differential Equation -With the assumption of irrotational motion and an incompressible fluid, a velocity potential exists which should satisfy the continuity equation: The divergence of a gradient leads to the Laplace equation: Substituting these velocities into the irrotationality condition again yields the Laplace equation: This equation must hold on the fluid But,if the motion had been rotational then we have w vorticidad (capacidad de rotación de un fluido) BOUNDARIES
Kinematic Free Surface Boundary Condition (KFSBC). The free surface of a wave can be described as F(x, y, z, t ) = z - q(x, y, t ) = 0, where q(x, y, t ) is the displacement of the free
surface about the horizontal plane, z = 0. The kinematic boundary condition at the free surface is
Carrying out the dot product yields Thus for cases in which surface tension forces are important, the dynamic free surface boundary condition is modified to SUMMARY PAGE 51
Dynamic free surface BC Since by our definition r] will have a zero spatial and temporal mean, C(t) = 0.5The terms within the brackets are constant; therefore, r] is given as a constant times periodic terms in space and time plus a function of time. We can rewrite r] as The velocity potential is now Kinematic free surface boundary condition The remaining free surface boundary condition will be utilized to establish the relationship between sigma and k. Noting that by definition a propagating wave will travel a distance of one wave length L, in one wave period T, and recalling that a = 2z/T and
k = 2n/L, it is clear that the speed of wave propagation C can be expressed
from Eq. (3.34) as The wave speed, or celerity, C, has been defined as C = LIT. Therefore,
Particle Displacements A water particle with a mean position of, say, (xi, zI) will be displaced by the wave-induced pressures and the instantaneous water particle position will be denoted as (xl + c, zI + 5), as shown in Figure 4.2. The displacement components (C, 5) of the water particle can be found by integrating the velocity with respect to time.
Pressure Refraction -Conditions -Contours straight and parallel to offshore -The longshore projection of wave number is a constant -Conservation of energy WAVE GROUPS
