HallsofIvy said:Well, not for a general g(r,t) but there are solutions for some specific, non-trivial functions.
The D'Alambert equation is a partial differential equation that describes the behavior of waves in a given medium. It is derived from the wave equation and is often used in physics and engineering to solve problems involving wave propagation.
The D'Alambert equation can be solved using separation of variables, where the solution is expressed as a product of two functions, one depending only on position ($\vec{r}$) and the other depending only on time ($t$). These two functions can then be solved separately using standard techniques.
The D'Alambert equation has a wide range of physical applications, including the study of sound waves, electromagnetic waves, and mechanical waves. It is also used in fields such as acoustics, optics, and fluid mechanics to model and analyze wave phenomena.
The D'Alambert equation is a linear equation, meaning that it only describes wave behavior in linear media. It also assumes that the medium is homogeneous and isotropic, and that there are no external forces acting on the waves. These limitations make it less applicable in certain real-world scenarios.
Yes, the D'Alambert equation can be extended to higher dimensions by adding more spatial variables. In three dimensions, for example, the equation becomes $\frac{1}{c^2}\frac{\partial^2\psi}{\partial t^2} - \nabla^2\psi = 0$, where $\nabla^2$ is the Laplacian operator. This extended version is often used in the study of three-dimensional waves, such as electromagnetic waves and seismic waves.