The tangent curve to Bessel functions?

juan's suggestion of the Hankel function seems great. Here is $J_0(x)$ and $u(x):=|J_0(x)+iY_0(x)|$.

Of course $u(x) \ge J_0(x)$ everywhere and $u(x) = |J_0(x)|$ at all the zeros of $Y_0(x)$. For asymptotic properties of $u(x)$, from Maple I get asymptotics $$ u(x) = \sqrt{\frac{2}{\pi x}}+O(x^{-3/2})\qquad\text{as } x \to \infty $$ which explains the good fit in the picture of the OP.

For example, At the zero $x_0 = 19.64130970$ of $Y_0(x)$, we have $$ J_0(x_0) = u(x_0) = 0.180005\qquad\text{and}\qquad \sqrt{\frac{2}{\pi x_0}} = 0.180034 , $$ a very close fit.