k-Wave User Forum » Topic: Understanding the derivation of the nonlinear terms
http://www.k-wave.org/forum/topic/understanding-the-derivation-of-the-nonlinear-terms
Support for the k-Wave MATLAB toolboxen-USThu, 08 Aug 2024 19:56:07 +0000http://bbpress.org/?v=1.0.2<![CDATA[Search]]>q
http://www.k-wave.org/forum/search.php
guillefix on "Understanding the derivation of the nonlinear terms"
http://www.k-wave.org/forum/topic/understanding-the-derivation-of-the-nonlinear-terms#post-9005
Tue, 16 Jan 2024 07:59:52 +0000guillefix9005@http://www.k-wave.org/forum/<p>I am trying to follow the derivation of the nonlinear terms in the wave equation used in k-wave, following the paper <a href="https://pubmed.ncbi.nlm.nih.gov/22712907/" rel="nofollow">https://pubmed.ncbi.nlm.nih.gov/22712907/</a></p>
<p>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. </p>
<p>I don't see how this assumption could hold.</p>
<p>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.</p>
<p>To elaborate on my confusion:</p>
<p>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??</p>
<p>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.</p>
<p>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??</p>
<p>Thank you for the help!:>
</p>