Chirp with linearly changing frequency and amplitude?

Question

A linear chirp

or linearly swept sine is a signal in which the frequency changes linearly with time:

the starting frequency changes into the ending frequency over time at a rate of:

and is the amount of time it takes. Its instantaneous frequency at point t is:

It looks like this:

Is there a similar equation where you can make the amplitude also change with time, independently of the changing frequency? It would have this shape approximately:

Answer 1

You simply have a time-dependent amplitude function $A(t)>0$, as already pointed out in a comment:

$$x(t)=A(t)\sin(\omega t+\phi)$$

The function $A(t)$ can have any form as long as it is positive (because it must be an amplitude). If you want it to decrease linearly from value $A_0>0$ at time $t_0$ to $A_1$ ($0<A_1<A_0$) at time $t_1$, simply define it as

$$A(t)=\frac{A_1-A_0}{t_1-t_0}(t-t_0)+A_0$$