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Engineering › Fluid Mechanics › Waves › Analytic ›

Stokes

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Stokes 5th order wave theory, which computes the surface profile and kinematics

Controller: will

Contents

Interface
Class Stokes
Stokes Eta
Stokes Kin
Stokes Surface Kin
Stokes MaxVel
Stokes Component
Page Comments

Interface

C++

Overview

The Stokes wave is an analytical model for regular (each wave looks identical), steady (the wave form doesn't change with time) and unidirectional (all travelling in the same direction) waves.

In such an environment, the Stoke's solution provides an accurate description of both the surface elevation and all the underlying kinematics. In particular it is the first model to demonstrate that:

the wave velocity increases slightly with increase of wave steepness.
the wave profile is more peaked at the crests with wide, less shallow, troughs.
the trajectories of water particles are not closed, leading to a mass transport in the direction of the waves, often called Stoke's drift.
the waves break when the wave crest angle reached $\inline 120^\circ$ .

In forming this solutions, Stokes (1874) assumed the fluid was inviscid and therefore irrotational, and also incompressible. Using the mathematical technique of series expansions (also called Stokes' expansion) he was able to derive a solution to the boundary conditions.

Class Stokes

This Stokes class provides a rapid mechanism for computing the surface elevation and internal kinematics within a wave. It is optimised on the assumption that many requests will be made for different spatial and temporal locations, where as the wave properties and water depth will be varied less frequently.

Example 1

To following example outputs the surface elevation and then the velocities beneath the central wave:

#include <codecogs/engineering/fluid_mechanics/waves/analytic/stokes.h>
#include <stdio.h>
 
int main()
{
  double Length=200;   // Wave length [m]
  double Amplitude=15; // Wave Height [m]
  double Depth=30;     // Depth [m]
  double time=0;       // Time [seconds]
  Engineering::Waves::Analytic::Stokes water(2*pi()/Length, amplitude, 0, depth);
 
  for(double x=-200;x<=200;x+=10)
    printf("\n Eta(%lf)=%lf",x, water.getEta(x, time);
 
  double eta=water.getEta(0,0); 
  for(double z=eta;z>-30;z--)
    printf("\n Phi(%lf)=%lf\tu(%lf)=%lf\tv(%lf)=%lf",
        z, water.getKin(0,z,time), 
        z, water.getKin(0,z,time,ddx), 
        z, water.getKin(0,z,time,ddz);
}

Authors

William Bateman (April 2010)

Source Code

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Members of Stokes

Stokes

Stokes(	double	`k`
	double	`amplitude`
	double	`phase` `= 0`
	double	`depth` `= 0`
	double	`gravity` `= 9.8066`
	int	`sign` `= 1`	)[constructor]

Constructor for this class. Call the init(...) function, below, with the supplied sea conditions and depth

Init

voidinit(	double	`k`
	double	`amplitude`
	double	`phase` `= 0`
	double	`depth` `= 0`
	double	`gravity` `= 9.8066`
	int	`sign` `= 1`	)

Computes a number of constants for the given sea state, which allow rapid evaluation of water surface at different temporal and spatial locations.

k	Wave number of dominant wave component [ $\inline 2\pi/m$ ]
amplitude	Amplitude of the dominant wave component [m]
phase	Phase adjustment for the wave in radians
depth	Depth of water, from sea bed to still water level [m]
gravity	Acceleration due to gravity [ $\inline m/s^2$ ]
sign	The Direction the wave propagate in. 1=left to right. -1=right to left.

Component

doublecomponent( int i )

Returns the amplitude of underlying frequency components

i	the ith component from 1 to 5

GetEta

doublegetEta(	double	`x`
	double	`time`	)

Computes the surface elevation at the point $\inline (x,time)$ for the given Stokes 5th wave profile with wave-number(k) and an amplitude.

x	Spatial location at which to compute the elevation [m]
time	Temporal location at which to compute the elevation [sec]

Returns

The surface elevation about still water level [m]

GetKin

doublegetKin(	double	`x`
	double	`z`
	double	`time`
	Derivative	`derivative` `= norm`	)

Computes the kinematics at the point $\inline (x,z,time)$ for the given Stokes 5th wave profile with wave-number(k) and an amplitude.

The derivate term allows different kinematics to be returns:

Norm return Phi( $\inline \phi$ )
ddx, computes $\inline \frac{d\phi}{dx}$ or the horizontal velocity (u)
ddz, computer $\inline \frac{d\phi}{dz}$ or the vertical Velocity (v)

x	Spatial location at which to compute the kinematics[m]
z	Depth at which to compute the kinematics [m]
time	Temporal location at which to compute the elevation [sec]
derivative	The derivative of Phi to compute

Returns

Velocity Potential [ $\inline m^2/s$ ] or Horizontal/Vertical velocity potential [ $\inline m/s$ ]

Stokes Eta

doubleStokes_eta(	double	`k`
	double	`amplitude`
	double	`x`
	double	`time` `= 0`
	double	`depth` `= 0`	)

Return the surface elevation for the Stokes 5th wave at the spatial and temporal locations specified.

These function are used when you need a quick estimate of a value at a specific point. For repeated evaluation it is best to use the Stokes Class.

The following graph shows the surface profile for a wave of length 224m (or period=12s) and a fundamental amplitude of 15m.

There is an error with your graph parameters for Stokes_eta with options k=0.0279 amplitude=15 x=-224:224 .width=450

Error Message:Function Stokes_eta failed. Ensure that: Invalid C++