Determine the sinusoidal function that will fall entirely within the exponential envelope.

Question

How do I determine the sinusoidal function that will lie perfectly between the two exponential functions (envelope): $1+e^{\left(-2t\right)}$ and $1-e^{\left(-2t\right)}$?

Thank you in advance.

Answer 1

I assume you want a damped sinusoidal, of the form $$A=f(t)\sin(\omega t)+b$$ Then take $b$ to be the average of your envelope functions. In this case $b=1$. Choose $f(t)$ to be half the difference between the envelope functions, $$f(t)=e^{-2t}$$ So $$A(t)=e^{-2t}\sin(\omega t)+1$$ Notice that since $\sin(\omega t)$ is between $-1$ and $1$, $$1-e^{-2t}\le A(t)\le 1+e^{-2t}$$