Solutions of the wave equation in one dimension have the form $$u(x,t) = F(x-ct)+G(x+ct)$$ if $F$ and $G$ are sufficiently well behaved. If $G=0$, this is just the $F$ traveling the real line, so to say, keeping its shape.

Lets define the diameter of the support set of $u$ as $$d_u(t) = \sup\{|x-y|\; \mid u(t, x)\neq0 \,\wedge\, u(t, y)\neq0\}$$ i.e. $d_u$ is the maximum (supremum) of the distance between any two points in the support set of $u$ at some fixed time $t$. In the one-dimensional case described above, this value does not actually change with time, since $F$ keeps it shape and therefore its support set just shifts along the real line.

In particular there exists a $D\lt\infty$ such that $$d_u(t)\lt D\;\;\forall t.$$

Question: Can we have solutions $u(t, x, y, z)$ in three dimensions which fulfill such a time-independent bound on the diameter of the support set?

Neither spherical nor plane waves fit the bill, but maybe some clever combination is possible? Or is it easy to prove that this is not possible? (OK, $u=0$ is a solution, but as you guessed, I am interested in non-trivial solutions).

EDIT: would it help if the wave moves on a (curved) (smooth) manifold?


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