Imagine we have for every real direction $\mathbf n \in \mathbb R^3$ a plane wave, be it either quantum, eletric, magnetic, acoustic or in my case some sheets of paper I am holding up. Each of these plane waves propagates with speed $c(\mathbf n)$ and amplitude
$$
f(\mathbf n \cdot x - c(\mathbf n) t),
$$
where $f$ is a scalar function.
Now here comes the question: what would be the envelope of all these plane waves? Said in another way, if
$$
W(x,t) = \int_{\mathbf R^3} f(\mathbf n \cdot x - c(\mathbf n) t) d\mathbf n,
$$
then, for each $t$, what is the surface $\mathcal S$ where $W(x,t)$ will be locally larger, and by locally I mean
$$ x \in \mathcal S \implies W(x,t) = \max_{|s^\perp| <<1} W( s^\perp+ x,t), $$
where $s^\perp$ is in the orthogonal compliment of the plane tangent to $\mathcal S$ at the point $x$. The answer is independent of $f$, and I believe now that it involves some simple intuitive trick (if I could I would add a picture).
Physically, this is the surface where the plane waves "intersect" and create a focus. This question is important for Fourier Transforms (other transforms) and the solver will certainly gain dark intuitive powers regarding waves.