According to one of my old notebooks, I came up with a generalization of the d’Alembert operator between 2022-03-27 and 2022-05-20. I can’t say that it was well-motivated by a particular mathematical or physical problem. It was just some low-hanging fruit in terms of mathematical generalization.
First, let us start with the definition of the d’Alembert in rectangular coordinates:
Definition
The d’Alembert operator in rectangular coordinates is given by
\[\square \triangleq \frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \sum_{j=1}^3 \frac{\partial^2}{\partial x_j^2}\]where $c$ is the speed of light in some desired choice of units.
It is then clear that the following generalization allows for different orders of partial derivative in any number of dimensions in rectangular coordinates.
Definition
The generalized d’Alembert operator in rectangular coordinates is given by
\[\square_k \triangleq \frac{1}{\alpha^k} \frac{\partial^k}{\partial t^k} - \sum_{j=1}^n \frac{\partial^k}{\partial x_j^k}\]which can also be written as
\[\square_k \triangleq \frac{1}{\alpha^k} \frac{\partial^k}{\partial t^k} - \vec 1_n \cdot \left( \bigodot_{i=1}^{k} \nabla_{\vec x} \right)\]where $\alpha$ is a parameter , and $\bigodot$ is the element-wise product.
Mathematically the d’Alembert has a relationship to the curvature of manifolds, but its generalization will certainly have a less-direct relationship to curvature for $k \neq 2$.