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''wave operator, d'Alembertian''
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''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>
 
  
 +
The second-order differential operator that in Cartesian coordinates assumes the following form:
 +
$$
 +
\Box u \stackrel{\text{df}}{=} \Delta u - \frac{1}{c^{2}} \frac{\partial^{2} u}{\partial t^{2}},
 +
$$
 +
where $ \Delta $ is the Laplace operator and $ c $ is a constant. Its form in spherical coordinates is:
 +
$$
 +
\Box u =
 +
\frac{1}{r^{2}} \frac{\partial}{\partial r} \left[ r^{2} \frac{\partial u}{\partial r} \right] +
 +
\frac{1}{r^{2} \sin(\theta)} \frac{\partial}{\partial \theta} \left[ \sin(\theta) \frac{\partial u}{\partial \theta} \right] +
 +
\frac{1}{r^{2} {\sin^{2}}(\theta)} \frac{\partial^{2} u}{\partial \phi^{2}} -
 +
\frac{1}{c^{2}} \frac{\partial^{2} u}{\partial t^{2}};
 +
$$
 
in cylindrical coordinates:
 
in cylindrical coordinates:
 
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$$
<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>
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\Box u =
 
+
\frac{1}{\rho} \frac{\partial}{\partial \rho} \left[ \rho \frac{\partial u}{\partial \rho} \right] +
 +
\frac{1}{\rho^{2}} \frac{\partial^{2} u}{\partial \phi^{2}} +
 +
\frac{\partial^{2} u}{\partial z^{2}} -
 +
\frac{1}{c^{2}} \frac{\partial^{2} u}{\partial t^{2}};
 +
$$
 
in general curvilinear coordinates:
 
in general curvilinear coordinates:
 +
$$
 +
\Box u = \frac{1}{\sqrt{- g}} \frac{\partial}{\partial x^{\nu}} \left[ \sqrt{- g} g^{\mu \nu} \frac{\partial u}{\partial x^{\mu}} \right],
 +
$$
 +
where $ g $ is the determinant of the matrix $ [g^{\mu \nu}] $ formed from the coefficients of the metric tensor $ g^{\mu \nu} $.
  
<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>
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Named after J. d’Alembert (1747), who considered its simplest form when solving the one-dimensional wave equation.
  
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" />.
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====Comments====
  
Named after J. d'Alembert (1747), who considered its simplest form when solving the one-dimensional wave equation.
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In the last equation above, the Einstein [[summation convention]] applies to the right-hand side (i.e., there is a summation involved over all indices $ \mu,\nu $).
  
 +
====References====
  
 
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<table>
====Comments====
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<TR><TD valign="top">[a1]</TD><TD valign="top">
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" />).
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R. Courant, D. Hilbert, “Methods of mathematical physics. Partial differential equations”, '''2''', Interscience (1965). (Translated from German)</TD></TR>
 
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<TR><TD valign="top">[a2]</TD><TD valign="top">
====References====
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F. John, “Partial differential equations”, Springer (1968).</TD></TR>
<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>
+
</table>

Latest revision as of 00:49, 15 December 2016

wave operator, d’Alembertian

The second-order differential operator that in Cartesian coordinates assumes the following form: $$ \Box u \stackrel{\text{df}}{=} \Delta u - \frac{1}{c^{2}} \frac{\partial^{2} u}{\partial t^{2}}, $$ where $ \Delta $ is the Laplace operator and $ c $ is a constant. Its form in spherical coordinates is: $$ \Box u = \frac{1}{r^{2}} \frac{\partial}{\partial r} \left[ r^{2} \frac{\partial u}{\partial r} \right] + \frac{1}{r^{2} \sin(\theta)} \frac{\partial}{\partial \theta} \left[ \sin(\theta) \frac{\partial u}{\partial \theta} \right] + \frac{1}{r^{2} {\sin^{2}}(\theta)} \frac{\partial^{2} u}{\partial \phi^{2}} - \frac{1}{c^{2}} \frac{\partial^{2} u}{\partial t^{2}}; $$ in cylindrical coordinates: $$ \Box u = \frac{1}{\rho} \frac{\partial}{\partial \rho} \left[ \rho \frac{\partial u}{\partial \rho} \right] + \frac{1}{\rho^{2}} \frac{\partial^{2} u}{\partial \phi^{2}} + \frac{\partial^{2} u}{\partial z^{2}} - \frac{1}{c^{2}} \frac{\partial^{2} u}{\partial t^{2}}; $$ in general curvilinear coordinates: $$ \Box u = \frac{1}{\sqrt{- g}} \frac{\partial}{\partial x^{\nu}} \left[ \sqrt{- g} g^{\mu \nu} \frac{\partial u}{\partial x^{\mu}} \right], $$ where $ g $ is the determinant of the matrix $ [g^{\mu \nu}] $ formed from the coefficients of the metric tensor $ g^{\mu \nu} $.

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 indices $ \mu,\nu $).

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://encyclopediaofmath.org/index.php?title=D%27Alembert_operator&oldid=40010
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article