Take the wave equation, which is hyperbolic. \begin{equation*} \nabla^2 u - \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} = 0 \end{equation*}
Now take a Fourier transform in time and you recover \begin{equation*} \nabla^2 u + k^2u = 0 \end{equation*} where $k^2 = \frac{\omega^2}{c^2}$ and $u(x,y,z,t) \to \hat{u}(x,y,z,\omega) := u(x,y,z,\omega)$ via Fourier transform. Looking at the definition at https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation, this differential equation is elliptic ($B=0$, $A=C=1 \Rightarrow B^2-AC=-1<0$). This same article says "Elliptic equations have no real characteristic curves, curves along which it is not possible to eliminate at least one second derivative of $u$ from the conditions of the Cauchy problem."
Is there an intuitive explanation as to why the Fourier transform will move the wave equation which is hyperbolic into the Helmholtz equation which is elliptic? Especially in terms of the quote above about characteristic curves?