# D'Alembert operator

From Encyclopedia of Mathematics

*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.

#### Comments

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

#### References

[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: http://www.encyclopediaofmath.org/index.php?title=D%27Alembert_operator&oldid=37604

This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article