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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|wave equation]] with variable coefficients:
 
A quantity describing the rate of propagation of a wave process in dispersing media. Let the wave process be described by the [[Wave equation|wave equation]] with variable coefficients:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g0453001.png" /></td> </tr></table>
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g0453002.png" /></td> </tr></table>
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\frac{1}{c  ^ {2} ( z) }
 +
 
 +
u _ {tt} - u _ {xx} - u _ {zz}  = 0,
 +
$$
 +
 
 +
$$
 +
0  \leq  z  \langle  \infty ,\  - \infty  < < \infty ,\  c ( z)  \rangle  0.
 +
$$
  
 
The solutions sought satisfy the conditions
 
The solutions sought satisfy the conditions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g0453003.png" /></td> </tr></table>
+
$$
 +
\left . u \right | _ {z = 0 }  = 0,\ \
 +
u _ {z \rightarrow \infty }  \rightarrow  0 ,
 +
$$
  
 
and have the form
 
and have the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g0453004.png" /></td> </tr></table>
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$$
 +
= \
 +
e ^ {i \omega ( k) t - ikx }
 +
v ( z).
 +
$$
 +
 
 +
The function  $  v ( z) $
 +
should be a non-zero solution of the one-dimensional boundary value problem
 +
 
 +
$$
 +
v  ^ {\prime\prime} +
 +
\left (
 +
k  ^ {2} -
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g0453005.png" /> should be a non-zero solution of the one-dimensional boundary value problem
+
\frac{\omega  ^ {2} }{c  ^ {2} ( z) }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g0453006.png" /></td> </tr></table>
+
\right )
 +
= 0; \ \
 +
\left . v \right | _ {z = 0 }  = 0; \ \
 +
v _ {z \rightarrow \infty }  \rightarrow  0.
 +
$$
  
If, in a certain range of variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g0453007.png" />, there exists a finite number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g0453008.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g0453009.png" /> for which this problem has a non-zero solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g04530010.png" />, then the quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g04530011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g04530012.png" /> are said to be, respectively, the phase and group velocities of the wave
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g04530013.png" /></td> </tr></table>
+
$$
 +
u _ {j}  = \
 +
e ^ {i \omega ( k) t - ikx }
 +
v _ {j} ( z).
 +
$$
  
 
The two velocities are related by the Rayleigh formula:
 
The two velocities are related by the Rayleigh formula:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g04530014.png" /></td> </tr></table>
+
$$
 +
= V -
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045300/g04530015.png" /> is the wave-length.
+
\frac{\lambda dV }{d \lambda }
 +
,
 +
$$
 +
 
 +
where $  \lambda $
 +
is the wave-length.
  
 
====References====
 
====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></table>
 
<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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Brillouin,  "Les tenseur en mécanique et en élasticité" , Masson  (1949)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Brillouin,  "Les tenseur en mécanique et en élasticité" , Masson  (1949)</TD></TR></table>

Revision as of 19:42, 5 June 2020


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)

Comments

References

[a1] L. Brillouin, "Les tenseur en mécanique et en élasticité" , Masson (1949)
How to Cite This Entry:
Group velocity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Group_velocity&oldid=47145
This article was adapted from an original article by V.M. Babich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article