Everyone knows the d'Alembert wave operator acting on a scalar function $\Psi(\boldsymbol{r},t)$ as:
$$ ( \partial_t^2/c^2 - \Delta ) \Psi(\boldsymbol{r},t) = 0 $$
The homogeneous solution in 3 dimensions ($\boldsymbol{r}$ is the position vector) is:
$$ \Psi = \frac{1}{r} F(r \pm c t) $$
with $F$ an arbitrary function. We can write the wave equation using the d'Alembert-box operator:
$$ \Box = \partial_t^2/c^2 - \Delta $$
Now let us consider this equation:
$$ ( \partial_t^2/c^2 + \Delta ) \Psi_s(\boldsymbol{r},t) = 0 $$
In this case, the 3d homogeneous solution should become:
$$ \Psi_d = \frac{1}{r} F(r \pm i c t) $$
An example of function $F$ could be $F(x) = e^{-k x}$, $k$ constant (wave vector like) and so we could have:
$$ \Psi_d \propto \frac{1}{r} e^{- k (r + i c t) } $$
Substracting the complex conjugate solution we obtain:
$$ \Psi_d \propto \frac{1}{r} e^{- k r } \sin(k c t) $$
That is a kind of standing wave (it oscillates but at the same location, no propagation). So if we define $\widehat{L}$ as:
$$ \widehat{L} = \partial_t^2/c^2 + \Delta $$
Can we talk about a standing wave operator ? Do the mathematicians already encounter this form of operator (related to physics or other fields)?
Thank you for reading.
Hint
Replacing $t$ by $i t$ transforms also the $\Box$ operator into $\widehat{L}$