D'Alembert operator

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).