Generalizing the pulse in Newton's derivation of the speed of sound I'm trying to understand this derivation of the speed of sound (its square equals the derivative of the pressure with respect to density), originally by Newton, I understand:
http://www.mathpages.com/home/kmath109/kmath109.htm
Here we make the assumption in the beginning that the sound pulse is of rectangular form. I understand the rest of the derivation.
I showed this to another person I know, and they questioned why would it be reasonable to believe that this result applies to all sound, not just a rectangular pulse. I found myself unable to provide an answer, other than that it seems reasonable to assume all disturbances travel at the same speed.
So, why is it that we can accept this derivation assuming a rectangular pulse and then conclude that it applies for all shapes of sound waves?
EDIT: It seems strange that we can conclude that the square of the speed equals the derivative of the pressure wrt. density, even though we used a rectangular pulse, a "discontinuous" wave. I've seen this identity used elsewhere with no restrictions to what kind of sound we are dealing with (what shape the wave is), such as in many fluid dynamics texts (physics of jet engines for example).
 A: The first thing to consider about the derivation presented is the edge of the pulse they chose - it doesn't go to or from zero (rather between two arbitrary pressures $p_1$ and $p_0$), and that even though this boundary represents decreasing pressure, it could just as well represent increasing pressure. Loosely speaking, the area under the curve of any function can be approximated as a series of thin rectangular boxes, which is the notion of Riemann sums. Take a look at the image on that page:

You can see that the rough outline of the curve can be traced using the tops of the boxes; the whole idea of the differential in calculus is that you can find a good approximation for exact solutions by letting the widths of these boxes become very small (3blue1brown has a very good explanation of this concept in his video series The Essence of Calculus). 
Thus, an arbitrary curve can be formed by making a pulse train of sorts (or rather, a series of steps), where the width of the pulse is assumed to be very small, and with each pulse scaled up according to the value of the function in question at that point. The derivation you provided could then be applied to each individual rise/fall.
