D'Alembert operator

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wave operator, d'Alembertian

The second-order differential operator which in Cartesian coordinates assumes the following form:

where is the Laplace operator and is a constant. Its form in spherical coordinates is:

in cylindrical coordinates:

in general curvilinear coordinates:

where is the determinant of the matrix formed from the coefficients of the metric tensor .

Named after J. d'Alembert (1747), who considered its simplest form when solving the one-dimensional wave equation.


In the last equation above, the Einstein (summation) convention applies to the right-hand side (i.e. there is a summation involved over all ).


[a1] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[a2] F. John, "Partial differential equations" , Springer (1968)
How to Cite This Entry:
D'Alembert operator. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article