This operator is called the d'Alembert operator.

(In the above equation, the square represents the D'Alembert operator.)

The square of is the Four-Laplacian, which is called the d'Alembert operator:

The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds.

The D'Alembert operator, which first arose in D'Alembert's analysis of vibrating strings, plays an important role in modern theoretical physics.

In the Minkowski space the Laplace-Beltrami operator becomes the d'Alembert operator or d'Alembertian:

In the modern mathematics of special relativity, electromagnetism and wave theory, the d'Alembert operator is the Laplace operator of Minkowski space.

The left hand sides of each equation correspond to wave motion (the D'Alembert operator acting on the fields), while the right hand sides are the wave sources.

The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function.

For calculations in Minkowski space, the D'Alembert operator, also called the D'Alembertian, wave operator, or box operator is represented as , or as when not in conflict with the symbol for the Laplacian.

The D'Alembert operator is also known as the wave operator, because it is the differential operator appearing in the wave equations and it is also part of the Klein-Gordon equation, which reduces to the wave equation in the massless case.