Why does absorption cause seismic pulses to increase in length over distance?

Question

The specific question I'm trying to answer is "How does the progressive loss of higher frequencies in a propagating seismic pulse lead to an increase in pulse length?"

I understand how the higher frequencies attenuate faster than lower frequencies (through the absorption constant), thus resulting in a change of shape but not so much how this increases the pulse length?

I've been researching for a while and have come across multiple ideas including:

1) I've read that in non-porous media velocity = freq x wavelength, as the frequency components reduce in value, the wavelength would increase to balance this? Though I'm not sure I can assume the media is non-porous.

2) I've been reading about wave dispersion which discusses how higher frequency components move faster than lower frequency components in dispersive media, thus the wavelength increases. My issue with this though is that the question specifically refers to the loss of higher frequencies influencing the pulse length, and just talking about different frequencies having different velocities doesn't seem to cut it.

3) Geometrical spreading could be increasing the wavelength? Though again this doesn't really relate to attenuation as the question is asking.

Any help letting me know what I'm missing here would be fantastic. This question has been bugging me for a couple of days now.

Cheers

Answer 1

Consider the initial pulse - it is of finite and short duration. It contains a number of frequency components, with higher and higher frequencies required for shorter and shorter initial pulses. This can be seen by taking the Fourier transform of your initial pulse. As the pulse propagates and the higher frequencies are preferentially absorbed, you can no longer continue to have as short of a pulse - the frequency spectrum does not allow it. Take the Fourier transform generated above, chop off the high frequency parts, and transform back - your pulse will be longer.