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I was told the sound wave loudness of a wave is proportional to the amplitude^2 .

This is true to perfect waves.

However, most sound waves in nature have compact base and are not of infinity length.

Which means, what would be the average loudness or the loudness of a sound wave that does not have a certain ampiltude. Maybe somthing like:

f(X) =

|X|*sin(X) if 0<X<1

(2-|X|)*sin(X) if 1<X<2

And 0 everywhere else.

Will it be integral of |f(X)|/2 ?

Thanks in advance.