What is the best way, in hardware, to amplify a decaying signal to a signal of uniform amplitude, as shown in the picture below? Does this amplification scheme have a name? I'm working in the audio range.
You're reinventing the guitar compressor (but that's fine).
As the commenters have noted it's an automatic gain control circuit that's required and they're used for many applications particularly to limit the signal level where exceeding the maximum will cause distortion by clipping of the audio.
You will come across terms such as:
Most of these devices work by converting the input signal into a DC voltage and using this to control the gain of an amplifier stage. Increasing control voltage results in lower gain. Compromises have to be made between rapid response to sudden changes in level - very significant in guitar as the strings have very high attack when plucked - and not so rapid that every cycle of a bass note causes the gain to change.