group celerity

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

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Contents

Interface
Group Celerity K
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Interface

C++

Group Celerity K

doublegroup_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 is 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:

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

(2)

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

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

(3)

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

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

(4)

Using this second form, we can plot as an example the relationship between Cg and wave-number, k :

There is an error with your graph parameters for group_celerity_k with options k=0:4 depth=1:10:4

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

Example 1

#include <stdio.h>
#include <codecogs/engineering/waves/group_celerity.h>
using namespace Engineering::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
??

Parameters

k	wave-number defines as ( $\inline 2 \pi/m$ ). [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/s²). [m/s²]

Returns

wave-frequency ( $\inline 2 \pi/s$ ). [rad/s]

Authors

Will Bateman (February 2005)

Source Code

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