Weakly Nonlinear Wave Theory for Periodic Waves (Stokes Expansion)
The solution for Stokes waves is valid in deep or intermediate water depth. It is assumed that the wave steepness is much smaller than one.
where is the wavenumber and is the water depth which is assumed constant.
- Nondimensional Variables
where and are dimensional variables and and are corresponding nondimensional variables.
Nondimensional Governing Equation & Boundary Conditions
where and stand for gradient and horizontal gradient, respectively.
Perturbation (Stokes Expansion)
Assuming the wave train is weakly nonlinear , its potential and elevation can be perturbed in the order of .
Using the Taylor expansion, the free-surface boundary conditions (Equations(3.1.3) and (3.1.4) are expanded at the still water level . Then we substitute perturbation forms of potential and elevation into the Laplace Equation, bottom and free-surface boundary conditions. The equations are sorted and grouped according to the order in wave steepness . The governing equations for order solutions is given by:
where the and can be derived in terms of the solutions for the potential and elevation of order or lower. Therefore, the above hierarchy equations must be solved sequentially from lower to higher order until the required accuracy is reached. To derive the third-order solution for a Stokes wave train, it is adequate to truncate the equations at .
Up to and are given below.
- Solving the non-dimensional Equations from lower order to higher order for the non-dimensional solutions ( wave advances in the x-direction ).
- The non-dimensional solutions are then transferred back to the dimensional form.
Nonlinear Dispersion Relation:
For the fast convergence of the perturbed coefficient, , must be much smaller than unity, which is consistent with weakly nonlinear assumption. However, when the ratio of depth to wave length is small, the Stokes perturbation may not be valid.
is the ratio of the potential magnitude of second-order to that of first order solution at .
For fast convergence, should be . This is true when . When , we have: , hence
may be much greater than unity sell number
For , then .
A few striking features of a nonlinear wave train can be described for the above equation:
- The crests are steeper and troughs are flatter; (see applet (Nonlinear Wave Surface)).
- Phase velocity increases with the increase in wave steepness.
- Non-closed trajectories of particles movement. (see applet (N-Trajectory)).
- Nonlinear wave characteristics (up to 2nd order).
Wave advancing in the x-direction
Denoting the mean position of a particle by , and its instantaneous displacement from the mean position by , the Lagrangian velocities of the particle are hence and , they are related to the Eulurian velocities through a Taylor Expansion:
where and are first- and second-order horizontal and vertical velocities.
We intend to compute up to second order in wave steepness
where superscripts stand for orders and overbar denotes a secular term. At lending-order, the solution is the same as that in LWT,
The leading-order trajectory of a particle is an ellipse of the center at and a major-axis and minor-axis .
The second-order solutions for the displacement are calculated by integrating the related second-order lagrangian velocities.
The secular term in the horizontal displacement indicates the particles will continuously move in the wave direction. Hence, the trajectory of a particle is no longer an ellipse. Because the horizontal mean position of a particle is not fixed at but change with time, we re-define the horizontal mean position by
Correspondingly, the displacement with respect to the instantaneous mean position is given by,<center>
The trajectory of a particle based on the solution is plotted in Applet (N-trajectory). The time average Lagrangian velocity of a particle is equal to the derivative of the secular term with respect to time.
The integral of the average Lagrangian velocity with respect to water depth renders the average mass flux induced by a periodic wave train over a unit width
which is consistent with the result derived using Eulurian approach.
Using the Bernoulli equation, dynamic pressure head induced by a periodic wave train can be calculated up to second-order,
Radiation stress: defined as the time average of excess quasi momentum flux due to the presence of a periodic wave train.
Up to second order, a wave train advancing in the x-axis,
, where , the energy density.
|In deep water||In shallow water|
In the case of a wave train having an angle, with respect to the x-axis,
This article is based on the lecture notes of Dr. Jun Zhang and the original article can be found here