In this context, a null field means a field constructed of planar waves $e^{I k_{\mu} x^{\mu}}$ that satisfy the null condition $k_{\mu} k^{\mu} = 0 \implies c^2 k^2 = \omega^2 $
Suppose we have a scalar field $V$ (which can be acoustic pressure, or a scalar electric potential) that is a solution of the wave equation
$$\Box V(x,y,z,t) = 0$$
I am wondering if a fixed (non-oscillating) minimum (or maximum) of field potential $\textbf{r}_0=(r_x,r_y,r_z)$ can be constructed with incoming waveforms on the boundary, or if it can be shown mathematically that is not possible
The Khirchhoff integral theorem provides certain ability to control fields inside a volume with sources on the boundary (holophony and acoustic levitation are based on this principle). But I am not sure what are the general restrictions that have the waveforms, and how those restrictions evolve with time
I tried to build an argument assuming that the waveform is made by finite Fourier elements (zero frequency / constant terms are not present as they are physically irrelevant )
$$f(x,t)=\sum_{k,\omega} \hat{f}(k,\omega) e^{-I(kx-\omega t)}$$
Assume that $x_0$ is a fixed point with $f(x_0,t) \ne 0$, $f_x(x_0,t) =0$, $f_{xx}(x_0,t)>0$, and finally that
$$\frac{\partial^{n} f(x,t)}{\partial t^{n-1} \partial x} \Bigg|_{x=x_0} = 0 $$
for all $n > 1$. This last condition ensures that the point stays a minimum over time. If these conditions are not satisfied exactly, $x_0$ will be a stable minimum only over some amount of time.
Now the part of the argument that leaves me unsure is to assume that since the function is only made of a finite sum of such planar waves, after taking enough derivatives I will have enough equations to prove that all coefficients should be zero, which would contradict some of the hypothesis $f(x_0,t) \ne 0$, $f_x(x_0,t) =0$.. etc.
Can this argument be improved? Also it doesn't say much if approximate fixed points can last for a period of time (probably yes, but I don't know how to prove that either)
A motivating example for this problem would be that, if you could create a stable minimum of electric potential with electromagnetic waves, you could sustain pseudo-electrostatic confinement potential for charged ions in a prescribed point that is otherwise devoid of matter, like those created with virtual grids in Fansworth fusors