CodeCogs

Computes the surface profile and kinematics of a Stokes 5th wave

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Interface

#include <codecogs/engineering/fluid_mechanics/waves/analytic/stokes.h>

using namespace Engineering::Fluid_Mechanics::Waves::Analytic;

Stokes()
Stokes(double k, double amplitude, double phase=0, double depth=0,double gravity=9.8066, int sign=1)
void init(double k, double amplitude, double phase=0, double depth=0,double gravity=9.8066, int sign=1)
double getEta(double x, double time)
double getKin(double x, double z, double time, Derivative derivative=norm)
double getMaxVel()
double Stokes_eta(double k, double amplitude, double x, double time=0, double depth=0)
double Stokes_kin(double k, double amplitude, double x, double z, double time=0, Derivative derivative=norm, double depth=0)
double Stokes_surface_kin(double k, double amplitude, double x, double time=0, Derivative derivative=norm, double depth=0)
double Stokes_maxVel(double k, double amplitude)

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 $120^\circ$ .

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.

Members of Stokes

CLASS METHOD

Stokes

Default constructor

CLASS METHOD

Stokes

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

CLASS METHOD

init

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

Parameters

Wave number of dominant wave component [ $2\pi/m$ ]

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 [ $m/s^2$ ]

sign

The Direction the wave propagate in. 1=left to right. -1=right to left.

CLASS METHOD

getEta

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

Parameters

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]

CLASS METHOD

getKin

Computes the kinematics at the point $(x,z,time)$ 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( $\phi$ )
ddx, computes $\frac{d\phi}{dx}$ or the horizontal velocity (u)
ddz, computer $\frac{d\phi}{dz}$ or the vertical Velocity (v)

Parameters

Spatial location at which to compute the kinematics[m]

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 [ $m^2/s$ ] or Horizontal/Vertical velocity potential [ $m/s$ ]

CLASS METHOD

getMaxVel

Returns the maximum possible values at all spatial and temporal locations

CLASS METHOD

Stokes_eta

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.

\graph k=0.0279 amplitude=15 x=-224:224 .width=450

Parameters

Wave number [ $2\pi/m$ ]

amplitude

Wave amplitude [m]

Spatial location [m]

time

Temporal location [sec]

depth

Depth of water [m]

Author

William Bateman (April 2010)

CLASS METHOD

Stokes_kin

Computes the kinematics within a Stokes 5th wave:

The following example shows the horizontal velocity profile with depth, beneath the crest of a wave of length 224m (or period=12s) and a fundamental amplitude of 15m.

\graph k=0.0279 amplitude=15 z=-20:15 x=0 time=0 derivative=1 depth=50 .width=400 .rotate

Parameters

Wave number [ $2\pi/m$ ]

amplitude

Wave amplitude [m]

Spatial location [m]

Vertical location [m]

time

Temporal location [sec]

derivative

The derivative of Phi

depth

Depth of water [m]

Author

William Bateman (April 2010)

CLASS METHOD

Stokes_surface_kin

This function combines the previous functions to first work out the height of the wave at a specific location, and then to compute the kinematics at this this surface locations.

The following graph shows the surface profile for a wave of length 224m (or period=12s) and a fundamental amplitude of 12m. \graph k=0.0279 amplitude=12 x=-200:200 derivative=1 .width=500

Parameters

Wave number [ $2\pi/m$ ]

amplitude

Wave amplitude [m]

Spatial location [m]

time

Temporal location [sec]

derivative

The derivative of Phi

depth

Depth of water [m]

CLASS METHOD

Stokes_maxVel

Returns the maximum velocity anywhere within a stokes wave within deep water. For a regular solution this maximum will always be the horizontal wave velocity in the peak of the wave crest.

Example solutions: \graph k=0.01:0.03:3 amplitude=0:20 .width=500

Parameters

Wave number [ $2\pi/m$ ]

amplitude

Wave amplitude [m]