An explicit equation for a special damped sinusoid.

Question

I apologize if my question seems weird.

The below damped sinusoid can be described by the following equation:

$$y(t) = A e^{\lambda t} \sin(\omega t)$$

Is it possible to manipulate this equation to produce something close to any of the following variations? (these figures are cited from 1950s papers at which the authors claimed that there are equations that produce such figures, but did not provide any!)

Answer 1

Variant 1: $f(t) = A e^{\alpha (t - t_0)} \sin (b (t - t_0))$

Variant 2: $f(t) = A e^{-\alpha t} \sin (b t) e^{- \beta (t - t_0)}$

The latter isn't perfect, but your figure itself is a bit unclear.