# Proof that fixed points of a null field are zero

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 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

I tried to build an argument assuming that the waveform is made by finite Fourier elements

$$f(x,t)=\sum_{k,\omega} \hat{f}(k,\omega) e^{-I(kx-\omega t)}$$

Assume that $x_0$ is a fixed point with $f(x_0,t) \ne 0$, this means that

$$\frac{\partial^{n} f(x,t)}{\partial t^n} \Bigg|_{x=x_0} = 0$$

for all $n > 0$.

Now the part of the argument that leaves me unsure is to assume that since the function is only made of a finite sum, after taking enough derivatives I will have enough equations to prove that all coefficients should be zero, which would contradict the hypothesis $f(x_0,t) \ne 0$.

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.

• I would suggest to take this question over to math.SE as someone with a firm background in PDE theory might well be able to give a useful answer (in other words, this can easily be restated as a pure maths question, but you have to be more specific on your boundary conditions). – Sebastian Riese Dec 21 '15 at 23:04
• I thought about it, but since it is related to holophony and wavefield synthesis, I thought it would find more expertise on Physics.SE – diffeomorphism Dec 22 '15 at 16:27
• There is a simple counterexample: Just add an off-set $\tilde V = V + 1$. Then $\tilde V$ will be a solution of the equation (with other boundary conditions) and a static point $\vec r_0$ of $V$ with $V(\vec r_0, t) = 0$ for all $t$ will be a static point of $\tilde V$ with value 1. So the question is ill posed. Of course it is an interesting question, whether a point can be a minimum at all times in a field that is a solution of the wave equation (it is not even necessary that the minimum have a $t$-independent value). – Sebastian Riese Dec 24 '15 at 3:37
• I should add that the DC term of the Fourier expansion must be zero, since to synthetise a finite waveform with the Kirchhoff theorem, you need nonzero frequencies – diffeomorphism Dec 24 '15 at 22:14