How to Turn the 555 Timer Output (Astable Mode) into a Decaying Square Wave?
What you want to do is to amplitude modulate a square wave.  Fortunately, this is easy to do with square waves because you can amplify them arbitrarily and then clip at a adjustable level.  This doesn't work with other waveshapes because the result is clipped tops and bottoms, so you end up with a square wave.  However, when you start with a square wave, that is fine.

Here is a simple circuit that exploits this effect:



The transistor will switch on and off according to the square wave.  The amplitude of the signal applied to the speaker is a function of the volume adjustment voltage, fed in at top.

R2 and D1 perform two functions.  First, they give the inductive flyback current a place to go when Q1 abruptly shuts off.  Without that, Q1 could get fried.  Second, R2 causes the current decay when Q1 is shut off to have about the same profile as the current rise when Q1 is turned on.  This isn't necessary to get sound out of the speaker, but will make the sound a little louder than if R2 was not there (replaced by a short).