View source for D'Alembert operator

''wave operator, d'Alembertian''

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d0300801.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d0300802.png" /> is the Laplace operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d0300803.png" /> is a constant. Its form in spherical coordinates is:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d0300804.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d0300805.png" /></td> </tr></table>

in cylindrical coordinates:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d0300806.png" /></td> </tr></table>

in general curvilinear coordinates:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d0300807.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d0300808.png" /> is the determinant of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d0300809.png" /> formed from the coefficients of the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d03008010.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030080/d03008011.png" />).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. John,   "Partial differential equations" , Springer  (1968)</TD></TR></table>