group_celerity — Waves

The velocity of a wave group (also known as 'group celerity')

Interface

#include <codecogs/engineering/fluid_mechanics/waves/group_celerity.h>

using namespace Engineering::Fluid_Mechanics::Waves;

double group_celerity_k(double k, double depth=0, double gravity=9.8066)

Within a train of waves the movement of energy is usually slower than that of the waves. As such, if you study a short train of waves they will develop at the back of the group, travel forward through the group and disappear at the front of the group.

[DIAGRAM]

The definition of group velocity is:

$$C_g = \frac{\partial \omega}{\partial k}$$

(1)

In deep water, where $\omega^2=gk$, group velocity is merely

$$C_g = \frac{1}{2}C$$

(2)

While in shallow water the form is a little more complicated, i.e.

$$C_g = \frac{g}{2 \omega} \left [ k d\, sech^2(k d) + tanh(k d) \right ]$$

(3)

Using this second form, we can plot as an example the relationship between Cg and wave-number, k : \graph k=0:4 depth=1:10:4

Example 1

#include <stdio.h>
#include <codecogs/engineering/fluid_mechanics/waves/group_celerity.h>
using namespace Engineering::Fluid_Mechanics::Waves;

int main()
{
  printf("   k         Cg ");
  for(double k=0.01; k<1;k+=0.1)
  {
    printf("\n %.6lf", k);
    double w=group_celerity_k(k,2);
    printf("  %.3lf", w);
  }
}

Output:

k         Cg
 0.010000  4.428
 0.110000  4.324
 0.210000  4.072
 0.310000  3.723
 0.410000  3.337
 0.510000  2.962
 0.610000  2.624
 0.710000  2.336
 0.810000  2.098
 0.910000  1.905

Parameters

wave-number, defined as ($2 \pi/m$). [rad/m]

depth

the depth of the water to mean sea level. A value of zero or less corresponds to deep water. [m]

gravity

acceleration due to gravity (default 9.8066 m/s\^2). [m/s\^2]

Returns

wave group velocity (group celerity). [m/s]

Interactive Calculator

depth

gravity

Result –