Why does loudness of sound vary as square of amplitude? Why not cube of amplitude or just amplitude?
You obtain this result on the mathematical derivation of the intensity.
Edit : This is for the intensity of the sound wave. The loudness(in dB), is measured as β = log(I/I0)dB, where I0 = 10-12W/m2.
You seem to be confused between intensity and loudness
Loudness is a much more complicated thing, as it (usually) refers to what the human ear perceives.
See https://en.wikipedia.org/wiki/Loudness for more information.
Separate names with a comma.