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Dispersion

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

C++
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K To W

 
doublek_to_wdoublek
doubledepth = 0
doublegravity = 9.8066 )
This function solves the linear dispersion equation, w^2 = g k tanh(k d), 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

Parameters

k(2\;\mathrm{\pi/m}) is wave-number. [rad/m]
depththe depth of the water to mean sea level. A value of zero or less corresponds to deep water. [m]
gravity(default 9.8066). [m/s2]

Returns

wave-frequency (2\;\mathrm{\pi/s}). [rad/s]
Source Code

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K To W

 
doublek_to_wdoublea
doublek
doubledepth = 0
doublegravity = 9.8066 )
This function solves a 2nd order dispersion relationship to more accurately 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:

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

Example:

Example - Second order Wave Number to Wave Frequency
Problem
Compute the second order Wave Number from Wave Frequency in water of depth 2m.
Workings
#include <stdio.h>
#include <codecogs/engineering/fluid_mechanics/waves/dispersion.h>
using namespace Engineering::Fluid_Mechanics::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=k_to_w(amp,k,2);
    printf("  %.3lf", w);
  }
}

Parameters

aamplitude of component with wavenumber k. [m]
kwave-number of component (2\;\mathrm{\pi/m}). [rad/m]
depththe depth of the water to mean sea level. A value of zero or less corresponds to deep water. [m]
gravity(default 9.8066\;\mathrm{m/s^2}). [m/s2]

Returns

wave-frequency (2\;\mathrm{\pi/s}). [rad/s]
Source Code

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W To K

 
doublew_to_kdoublew
doubledepth = 0
doublegravity = 9.8066 )
Finds the wave-number associated with a particular wave-frequency, using this 1st order dispersion equation, w^2 = g k tanh(k d) to obtain wave-number from a given wave-frequency.

In Deep water, this equations reduces to w^2 = g k, 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 \partial \epsilon / \partial k.

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 w=1.5 only 4 iterations are needed, while 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 (2\pi / w) to the wave-length (2\pi/k) of a water wave. This is also shown graphically in the following figure:
\graph  w=0:4, depth=1:5:3, gravity=9.8066
The opposite of this function is dispersion_k

Example:

[metric]
Example - First order conversion of Wave Frequency to Wave Number
Problem
Compute Wave Frequency from Wave Number in water of depth 2m.
Workings
#include <stdio.h>
#include <codecogs/engineering/fluid_mechanics/waves/dispersion.h>
using namespace Engineering::Fluid_Mechanics::Waves;
 
int main()
{
  printf("   k         w  recalculated k");
  for(double k=0.01; k<1;k+=0.1)
  {
    printf("\n %.6lf", k);
    double w=k_to_w(k,2);
    double k2=k_to_k(w,2);
    printf("  %.3lf  %.6lf", w, k2);
  }
  return 0;
}
Solution
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

Example 1

Parameters

w(2\;\mathrm{\pi/s}) is wave-frequency, usually written using \omega (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 9.8066\;\mathrm{m/s^2}).

Returns

wave-number (2\;\mathrm{\pi/m}).
Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.