terms in equation: sloshing response to seismic waves

[kauri]

I'm using the equations from this research paper

https://www.iitk.ac.in/nicee/wcee/article/WCEE2012_0490.pdf

I'm hoping to use the linear analysis as shown in the paper.

What are the terms X and τ in this equation supposed to represent?

 

[WARose]

τ is a time. (Different than t.) It's the same notation used in Duhamel's integral. The X is most likely acceleration of a displacement.

 

[kauri]

Ok i'll do a bit of reading about Duhamel's integral thanks.
If you look at the governing equations section it actually explains that

, so ug is the acceleration; I'm not really sure how to go from my ug to X.