Pulse envelope - Knowlpedia

A pulse envelope is a positive function $P(v)$ used in a bound such as $|a(v)|\le C P(v)$ for a pulse amplitude. For a prescribed continuous net growth rate $g$ and reference point $v_0$ , one useful choice is

P(v)=\exp\left(\int_{v_0}^{v}g(s)\,ds\right).

Then $P(v_0)=1$ and $P'=gP$ . The exponential is positive, so ratios $P(v)/P(w)$ are well-defined.

Growth followed by decay

If $g$ is positive before $v_0$ and negative afterward, this envelope has a maximum at $v_0$ . In an amplitude equation $g$ can represent amplification minus viscous damping. Bounds for each derivative of $a$ must still be proved separately; an envelope inequality on values does not imply smoothness of $a/P$ or a zero extension at an endpoint.