As a sound burst attenuates does it get longer? I am trying to understand what a sound burst will look like after frequency dependent attenuation.
I come from engineering / signal processing not physics background.
If I am told as an engineer to mathematically attenuate high frequency I would apply a filter to the signal.  This results (typically) in a new signal which is longer in the time domain than the original.
Is this what happens, i.e. do physical waves, if experiencing frequency dependent attenuation, elongate in time; or is a FIR filter not a good approximation of physical attenuation? (by physical waves I do not necessarily in human audible range)


 A: Re. "Is this what happens to real sounds waves as they attenuate"
Yes: a pulse tends to become less oscillatory as it propagates through a medium that attenuates as a function of frequency and also gets stretched out in time. An FIR filter should be a good model of this.
For example if you convolve a single cycle of a sine wave with wave length l with a rectangular window of length L you will get a result of length l+L. (You can see this graphically on a sheet of paper by sliding he two functions past each other.) 
The rectangular window is a low pass filter with a sinc function frequency response (sin x over x).  It will also add a delay equal to 1/2 the window width.
See: Hamming "Digital filters" or Brigham "The fast Fourier transform".
A: No, a filter is not a good model for the attenuation of sound.
The attenuation with distance in free air is due the the $r^{-2}$ dependence of power. That factor affects all frequencies in the same way. There is also no dispersion, the velocity of sound does not depend on frequency.
In reality, of course, other things may happen. The signal may start to sound muffled, but that is primarily because frequency dependencies in absorption and reflection by walls, multiple propagation paths, etc.
