# Interal waves and the dispersion relation

Internal waves can exist in a stratified fluid. An interesting feature of these waves is the so-called dispersion relation between the angle of wave propagation (\theta), stratification strength (N^2) and the freqency of the wave (\omega), according to,

\displaystyle \omega = N^2 \cos(\theta).

The Navier-Stokes equations under the Boussinesq approximation are solved. In the centre of the domain an oscillating force exites the internal waves with a freqency corresponding to \theta = 45^o.

A pass for the 4th order solver

#include "nsf4t.h"
#include "view.h"
scalar b[], zeros[], * tracers = {b};
b[bottom] = dirichlet_vert_bottom;
b[top] = dirichlet_vert_top (L0);
vector av;

double omega;

int main() {
omega = sqrt(2)/2.;
periodic (left);
av.x = zeros;
av.y = b;
L0 = 30;
a  = av;
TOLERANCE = 1e-5;
DT = 0.2;
N = 128;
run();
}

event init (t = 0) {
foreach_vert()
b[] = y;
boundary ({b});
}

event accel (i++) {
foreach()
if (sq(x - L0/2) + sq(y - L0/2) < 1)
u.y[] += 0.1*dt*sin(omega*t); // Not accurate, but flexible...
}

event output (t += 0.5) {
scalar grb[];
foreach() {
grb[] = 0;
foreach_dimension()
grb[] += sq((b - b[-1])/(2*Delta));
grb[] = sqrt(grb[]);
}
output_ppm (grb, file = "grb.mp4", min = 0.8, max = 1.2, n = 300);
}

event stop (t = 75);