Steady-state wave equation

Question

I know that for the heat equation $u_t=\nabla^2u$, the steady-state condition $u_t=0$ suggests we need to solve $\nabla^2u=0$ (i.e. Laplace's equation), which yields the equilibrium temperature distribution after a considerable amount of time.

However, for the wave equation, say $u_{tt}=\nabla^2u$, if we consider when $u_{tt}$ is identically zero, what can we say about wave propagations here?

The only thing I found online about steady-state waves seem to mention something about elastic waves?

Answer 1

When $u_{tt}$ is identically zero we have a static situation. A static electric and/or magnetic field for example. Or a body under static stress. There are no waves.

Edit: Besides static situations, it is also possible linearly increasing or decreasing of a function. A good example is a rod subject to a tensile test, if the load is being increased linearly with the time.