Difference between revisions of "Wave vector"
From Encyclopedia of Mathematics
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| + | $#C+1 = 9 : ~/encyclopedia/old_files/data/W097/W.0907150 Wave vector | ||
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| − | + | The vector $ \mathbf k = ( k _ {1} \dots k _ {m} ) $ | |
| + | in the expression | ||
| − | + | $$ \tag{* } | |
| + | a \mathop{\rm exp} \left ( i \sum _ {j = 1 } ^ { m } | ||
| + | k _ {j} x _ {j} - i \omega t \right ) , | ||
| + | $$ | ||
| + | where $ a $ | ||
| + | and $ \omega $ | ||
| + | are constants and $ t $ | ||
| + | denotes time. | ||
| + | The usual physical interpretation of (*) is a plane wave of frequency $ \omega $, | ||
| + | propagating in the direction of the vector $ \mathbf k $ | ||
| + | and having wave-length $ \lambda = 2 \pi / | \mathbf k | $, | ||
| + | where $ | \mathbf k | = \sqrt {k _ {1} ^ {2} + \dots + k _ {m} ^ {2} } $. | ||
| + | Many homogeneous linear equations and systems of partial differential equations (including the more important equations of mathematical physics such as the [[Maxwell equations|Maxwell equations]] and the [[Wave equation|wave equation]]) have solutions in the form (*). | ||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. John, "Plane waves and spherical means applied to partial differential equations" , Interscience (1955)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I.M. Gel'fand, G.E. Shilov, "Generalized functions" , '''1. Properties and operations''' , Acad. Press (1964) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. John, "Plane waves and spherical means applied to partial differential equations" , Interscience (1955)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I.M. Gel'fand, G.E. Shilov, "Generalized functions" , '''1. Properties and operations''' , Acad. Press (1964) (Translated from Russian)</TD></TR></table> | ||
Latest revision as of 08:28, 6 June 2020
The vector $ \mathbf k = ( k _ {1} \dots k _ {m} ) $
in the expression
$$ \tag{* } a \mathop{\rm exp} \left ( i \sum _ {j = 1 } ^ { m } k _ {j} x _ {j} - i \omega t \right ) , $$
where $ a $ and $ \omega $ are constants and $ t $ denotes time.
The usual physical interpretation of (*) is a plane wave of frequency $ \omega $, propagating in the direction of the vector $ \mathbf k $ and having wave-length $ \lambda = 2 \pi / | \mathbf k | $, where $ | \mathbf k | = \sqrt {k _ {1} ^ {2} + \dots + k _ {m} ^ {2} } $. Many homogeneous linear equations and systems of partial differential equations (including the more important equations of mathematical physics such as the Maxwell equations and the wave equation) have solutions in the form (*).
Comments
References
| [a1] | F. John, "Plane waves and spherical means applied to partial differential equations" , Interscience (1955) |
| [a2] | I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 1. Properties and operations , Acad. Press (1964) (Translated from Russian) |
How to Cite This Entry:
Wave vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wave_vector&oldid=11958
Wave vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wave_vector&oldid=11958
This article was adapted from an original article by V.M. Babich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article