I am trying to follow the derivation of the nonlinear terms in the wave equation used in k-wave, following the paper https://pubmed.ncbi.nlm.nih.gov/22712907/

However, I don't understand the justification of equations (3) and (4) in that paper. I understand we are assuming that the acoustic density, pressure, etc are small. However, (3) and (4) seem to need the displacement field, and the *time since t_0, where t_0 is the time where the fluid is at equilibrium* to be small.

I don't see how this assumption could hold.

This doesn't seem to affect the derivation much, except for the term with displacement in the pressure-density relation. But that term matters for the eventual equation.

To elaborate on my confusion:

If we look at equation (4), that will only hold for very short times - surely shorter than we want to run our simulation, and therefore shorter than we want our equations to hold right??

But, this is ok, for equation (5). Equation (5) would still be valid for any finite time, if we have $\del{\rho}$ rather than $\del{\rho_0}$, and the displacement was small. However, we then see that the argument in the next line that $\hat{s(t_1)} - \hat{s(t_0)} = s$ and $\hat{s(t_1)} - \hat{s(t_0)} = p$, and that would change the equaitons very significantly.

So either a) For some reason, we only the equations to hold for short enough time than (4) is a good approximation, b) I am missing something about how this would extend for longer times, or c) I am missing something else??

Thank you for the help!:>