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)|
D'Alembert operator. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=D%27Alembert_operator&oldid=37604