Wave vector

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 (*).

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)