Uses a linear dispersion relationship to compute wave-frequency from wave-number

Contents

Interface Function Documentation Page Comments

Interface

#include <codecogs/engineering/waves/dispersion.h>

using namespace Engineering::Waves;

	double	dispersion_k (double k, double depth=0, double gravity=9.8066) Uses a linear dispersion relationship to compute wave-frequency from wave-number
	double	dispersion_k2 (double a, double k, double depth=0, double gravity=9.8066) Uses a 2nd order dispersion relationship, to obtain wave-frequency from wave-number
	double	dispersion_w (double w, double depth=0, double gravity=9.8066) Itartively solves for wave-number from wave-frequency.

Function Documentation

This function solves the linear dispersion equation, , to obtain wave-frequency from a given wave-number, using a very simple rearrangement where w is wave-frequency, k is wave-number and g is gravity. In deep water (represented with d<=0), this solution reduces to

The opposite of this function is dispersion_w

Example 1:

#include <stdio.h>
#include <codecogs/engineering/waves/dispersion.h>
using namespace Engineering::Waves;
 
int main()
{
  printf("   k         w ");
  for(double k=0.01; k<1;k+=0.1)
  {
    printf("\n %.6lf", k);
    double w=dispersion_k(k,2);
    printf("  %.3lf", w);
  }
}

Output:

k         w
0.010000  0.044
0.110000  0.483
0.210000  0.904
0.310000  1.294
0.410000  1.648
0.510000  1.962
0.610000  2.241
0.710000  2.489
0.810000  2.710
0.910000  2.910

Parameters:

k	() is wave-number. [rad/m]
depth	the depth of the water to mean sea level. A value of zero or less corresponds to deep water. [m]
gravity	(default 9.8066). [m/s^2]

Returns:

wave-frequency (). [rad/s]

This function solves a 2nd order dispersion relationship to more accurate compute the relationship between wave-frequency and wave-number for the specified wave amplitude a. The solution is based on work by Dalrymple, who derived:

where

In deep water(represented with d<=0), this solution reduces to As in the linear dispersion relationship.

Example 2:

#include <stdio.h>
#include <codecogs/engineering/waves/dispersion.h>
using namespace Engineering::Waves;
 
int main()
{
  double amp=3;  // 3 meters
  printf("   k         w ");
  for(double k=0.01; k<1;k+=0.1)
  {
    printf("\n %.6lf", k);
    double w=dispersion_k2(amp,k,2);
    printf("  %.3lf", w);
  }
}

Output:

k         w
????

Parameters:

a	amplitude of component with wavenumber k. [m]
k	wave-number of component (). [rad/m]
depth	the depth of the water to mean sea level. A value of zero or less corresponds to deep water. [m]
gravity	(default ). [m/s^2]

Returns:

wave-frequency (). [rad/s]

Finds the wave-number associated with a particular wave-frequency, using this 2nd order dispersion equation, to obtain wave-number from a given wave-frequency.

In Deep water, this equations reduces to , which can obviously solved directly.

For shallow water, an iterative approach must be used. For each iteration, we calculate the residual error: where w is wave-frequency, k is wave-number and g is gravity.

We then seek to minimise ε using the first derivative .

This convergence of this error function is fairly rapid, with poorest conversion, 8 iterations in water 2m deep (for 6dp precision) occurring when w is small i.e. w<=0.1. When \e w=1.5 only 4 iterations are needed, while \e w>3.5 needs only 2 iterations. Shallow water naturally requires more iteration. If your doing many calculation in either shallow water or very long wave periods (>30s), then you might want to consider fine tuning this function.

This relationship can me use to move from the wave-period () to the wave-length () of a water wave. This is also shown graphically in the following figure: The opposite of this function is dispersion_k

Example 3:

#include <stdio.h>
#include <codecogs/engineering/waves/dispersion.h>
using namespace Engineering::Waves;
 
int main()
{
  printf("   k         w  recalculated k");
  for(double k=0.01; k<1;k+=0.1)
  {
    printf("\n %.6lf", k);
    double w=dispersion_k(k,2);
    double k2=dispersion_w(w,2);
    printf("  %.3lf  %.6lf", w, k2);
  }
  return 0;
}

Output:

k         w  recalculated k
0.010000  0.044  0.010000
0.110000  0.483  0.110000
0.210000  0.904  0.210000
0.310000  1.294  0.310000
0.410000  1.648  0.410000
0.510000  1.962  0.510000
0.610000  2.241  0.610000
0.710000  2.489  0.710000
0.810000  2.710  0.810000
0.910000  2.910  0.910000

Parameters:

w	() is wave-frequency, usually written using (omega).
depth	(m) define the depth of the water to mean sea level. A value of zero or less corresponds to deep water.
gravity	(default ).

Returns:

wave-number ().