Derivation of group velocity equation

An impulse function at the origin in (x,z)-space is a superposition of Fourier components:  

$$\int \int \ e^{ +\,i\,k_x x \ +\ i\,k_z z } \ dk_x \ dk_z$$ (22)

Physics (and perhaps numerical analysis) leads to a dispersion relation that is a functional relation between $\omega$, k_x, and k_z, say, $\omega ( k_x , k_z )$. The most common example of such a dispersion relation is the scalar wave equation $\omega^2 = v^2 \, ( k_x^2 \ +\ k_z^2 )$.The solution to the equation is  

$$e^{ -\,i\, \omega t \ +\ i\, k_x x \ +\ i\, k_z z }$$ (23)

Integrating (23) over (k_x , k_z ) produces a monochromatic time function that at t = 0 is an impulse at (x,z) = (0,0). This expression at some very large time t is  

$$\int \int \ e^{ -\,i\,t\, [ \omega (k_x , k_z ) \ -\ k_x \ x / t \ -\ k_z \ z / t ] } \ dk_x \ dk_z$$ (24)

At t very large, the integrand is a very rapidly oscillating function of unit magnitude. Thus the integral will be nearly zero unless the quantity in square brackets is found to be nearly independent of k_x and k_z for some sizable area in (k_x , k_z )-space. Such a flat spot can be found in the same way that the maximum or minimum of any two dimensional function is found, by setting derivatives equal to zero. This analytical approach is known as the stationary phase method. It gives

$$\begin{eqnarray} 0 \ \ \ \ &=&\ \ \ \ { \partial \ \ \over \partial k_x } \ [\ \... ...\ \eq \ { \partial \omega \ \over \partial k_z } \ -\ {z \over t }\end{eqnarray}$$ (25) (26)

So, in conclusion, at time t the disturbances will be located at  

$$(x,z) \ \eq \ t \ \left( { \partial \omega \ \over \partial k_x } \ ,\ { \partial \omega \ \over \partial k_z } \, \right)$$ (27)

which justifies the definition of group velocity.

Now let us see how the left side of Figure 8 was calculated. The 15$^\circ$ dispersion relation was solved for $\omega$ and inserted into (27). The resulting (x,z) turned out to be a function of $k_x / \omega$.Trying all possible values of $k_x / \omega$ gave the curve.