I am looking for an envelope of the function $t\mapsto \mbox{sinc}^2(t)$, or at least an approximation of such an envelope. Is there a known envelope function?
Here's an illustration of $\mbox{sinc}^2(t) = \frac{\sin^2(\pi t)}{(\pi t)^2}$ (blue), and of one of its envelopes (red), which I drew by hand.
More generally, I was wondering, how does one calculate an envelope?
On
Only a single parameter curve can have a unique envelope obtained by $ c, p$ discriminant methods.
Your sketch includes a parameter which you have not identified, but implied.
For example a sine curve needs to be associted with parameter $c$ in $ y= A \sin c x, $ so that an elimination of parameter $c$ leads to envelope $y=\pm A.$
Following example is given for the curve $$ y=\frac{ \sin cx} {x},$$ the sinc squared curve could follow with the same procedure.
By partial differentition with $c$ one obtains
$$ \cos cx = 0 , \quad \sin cx = \pm 1, \rightarrow \, y=\frac{\pm1}{x} $$ which are hyperbolae curves are exactly sketched below.
An envelope is defined for a parametric family of functions and not for a single member. It is obtained by eliminating the parameter between the function equation and that obtained by differentiating over the parameter.
For example, consider the family
$$y=\frac{\sin^2\lambda t}{t^2}.$$ The aformentioned derivative is
$$0=\frac2t\sin\lambda t\cos\lambda t,$$ which implies $\sin^2\lambda t=0$ or $1$.
Hence the two envelopes
$$y=\frac1{t^2},\\y=0.$$