Group velocity

A quantity describing the rate of propagation of a wave process in dispersing media. Let the wave process be described by the wave equation with variable coefficients:

$$ \frac{1}{c ^ {2} ( z) } u _ {tt} - u _ {xx} - u _ {zz} = 0, $$

$$ 0 \leq z \langle \infty ,\ - \infty < x < \infty ,\ c ( z) \rangle 0. $$

The solutions sought satisfy the conditions

$$ \left . u \right | _ {z = 0 } = 0,\ \ u _ {z \rightarrow \infty } \rightarrow 0 , $$

and have the form

$$ u = \ e ^ {i \omega ( k) t - ikx } v ( z). $$

The function $ v ( z) $ should be a non-zero solution of the one-dimensional boundary value problem

$$ v ^ {\prime\prime} + \left ( k ^ {2} - \frac{\omega ^ {2} }{c ^ {2} ( z) } \right ) v = 0; \ \ \left . v \right | _ {z = 0 } = 0; \ \ v _ {z \rightarrow \infty } \rightarrow 0. $$

If, in a certain range of variation of $ k $, there exists a finite number of $ \omega _ {j} ( k), $ $ k = 1, 2 \dots $ for which this problem has a non-zero solution $ v _ {j} $, then the quantities $ V = \omega _ {j} ( k)/k $ and $ U = d \omega _ {j} /d k $ are said to be, respectively, the phase and group velocities of the wave

$$ u _ {j} = \ e ^ {i \omega ( k) t - ikx } v _ {j} ( z). $$

The two velocities are related by the Rayleigh formula:

$$ U = V - \frac{\lambda dV }{d \lambda } , $$

where $ \lambda $ is the wave-length.

References

[1]	L.I. Mandel'shtam, Complete works , 5 , Leningrad (1950) pp. 315–319; 419–425; 439–467 (In Russian)
[2]	G.S. Gorelik, "Oscillations and waves" , Moscow-Leningrad (1950) (In Russian)
[a1]	L. Brillouin, "Les tenseur en mécanique et en élasticité" , Masson (1949)