celerity — Waves

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

View versions (2)

Interface

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

using namespace Engineering::Fluid_Mechanics::Waves;

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

The velocity of a wave is defined by

$$c= \frac{\omega}{k}$$

(1)

where ω is the wave frequency and k is the wave-number.

Blank

Using a linear dispersion relationship, i.e. $w^2 = g k tanh(k d)$, we can derive a more useful expression for the wave velocity given a specific value of k, i.e.

$$c = \sqrt { \frac{g}{k} tanh(kd) }$$

(2)

In deep water (represented with d<=0), this solution reduces to

$$c = \sqrt { \frac{g}{k} }$$

(3)

Example 1

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

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

Output:

k         c
 0.010000  4.428
 0.110000  4.393
 0.210000  4.305
 0.310000  4.175
 0.410000  4.018
 0.510000  3.848
 0.610000  3.674
 0.710000  3.505
 0.810000  3.346
 0.910000  3.198

Parameters

wave-number, defined as (2π /m). [rad/m]

depth

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

gravity

(default 9.8066 m/s\^2). [m/s\^2]

Returns

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

Interactive Calculator

depth

gravity

Result –