In a pool of water, pushing my hand outwards faster seems to generate faster moving waves than pushing my hand slower. It seems the speed of the waves depends on the "suddenness" of the displacement.
I just realized I don't understand why one can't generate a "quicker" sound by some aggressive procedure, and on the other hand also don't understand how one might try to. I guess I just don't understand sound too well.
Is there an analogue the water phenomenon for sound? If not, why?
...the speed of the waves depends on the "suddenness" of the displacement.
This is two phenomena: that of Dispersion, where the speed of a wave depends on its frequency, and Nonlinearity where the speed of the wave depends on its amplitude.
A "more aggressive" thrust means both a faster onset (more higher frequency content of the wave) and more amplitude. Water waves are highly dispersive, as the Wiki article shows. The deep explanation really only comes through studying the equations carefully, so it's not trivial to understand. But the basis of the two pages I link is simple: one applies the Newtonian conservations laws of energy and momentum to all elements of the fluid at once. There's nothing more magical going on.
The best example of sound dispersion of know of is in the 1970s use of the slinky spring to produce the classic "laser blaster" sound in iconic films such as Star Wars. See Emilio Pisanty's great answer here, including Mathematica code for the sound generation. One taps the slinky, and the high frequency components of the shock arrive at the other end first, followed by the lower frequency ones. The result is the distinctive downwards-in-frequency "chirp" in the sound.
A well known example of nonlinear acoustics is the "Sonic Boom" radiating from a faster than sound object travelling through the air. Whereas sound dispersion tends to spread a sharp pulse out as it travels, some nonlinearities can concentrate it and keep it sharp and this is why the shock wave is not smoothed out. (The countervalent effects of dispersion and nonlinearity can lead to solitonic waves, although I'm not knowledgeable enough to know when they can happen in acoustics).
Nonlinearity also produces harmonics and distortion of sinusoidal sound waves. Nonlinearity is essential to the production of high quality sound by some musical instruments: I understand the acoustic nonlinearity of a pianoforte's frame and the intermodulation of chords that results is essential to the instrument's effervescent sound.
