View source for Group velocity

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