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group celerity

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

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:
$C_g&space;=&space;\frac{\partial&space;\omega}{\partial&space;k}$

In deep water, where $\inline&space;&space;\omega^2=gk$, group velocity is merely
$C_g&space;=&space;\frac{1}{2}C$
While in shallow water the form is a little more complicated, i.e.
$C_g&space;=&space;\frac{g}{2&space;\omega}&space;\left&space;[&space;k&space;sech^2(k&space;d)&space;+&space;tanh(k&space;d)&space;\right&space;]$

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 ($\inline&space;&space;2&space;\pi/m$). [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 ($\inline&space;&space;2&space;\pi/s$). [rad/s]

Authors

Will Bateman (February 2005)
Source Code

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