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I am dealing with an equation from nonlinear acoustics (Khokhlova-Zabolotskaya-Kuznetsov equation) where a strange term (for me as a mathematician) is used.

The equation looks like this $$ \frac{\partial}{\partial\theta} \left( \frac{\partial V}{\partial\theta} - V \frac{\partial V}{\partial\theta} \right) = const \left( \frac{\partial^2 V}{\partial\xi^2} + \frac{1}{\xi} \frac{\partial^2 V}{\partial\xi^2} \right) $$ Here $V:(\mathbb{R}^+)^3 \to \mathbb{R}$, $V=V(r,x,t)$ and $\xi = r/const$, $\theta = const(t-x/const)$, $\sigma=x/const$

The question is: what does expression $$ \frac{\partial V}{\partial\theta} = \frac{\partial V}{\partial (const (t-x/const) )}$$ mean? (of course one can just erase all $const$, they do not change anything)

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    $\begingroup$ From the Wikipedia article, look at the parameters of Burger's equation; the KZK equation is an augmentation of Burger's equation. The parameters are explained for Burger's. $\endgroup$ – Peter Diehr Apr 22 '16 at 19:33
  • $\begingroup$ @PeterDiehr do you know, can this equation be transformed to a usual PDE (without delay) like the Burger's one? $\endgroup$ – demitau Apr 23 '16 at 21:50
  • $\begingroup$ I don't think so; that's why it is always solved numerically, according to the article. $\endgroup$ – Peter Diehr Apr 23 '16 at 23:33
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The problem is that your point view is too "mathematical". No offence, but every acoustician would jump to the ceiling hearing "one can just erase all const, they do not change anything". Oh, they do $-$ very much! Since one of them is the sound speed... But I get it, you solve that as a mathematical problem and we are undoubtedly grateful for such people.

The $\theta$ is called retarded time and its often used in problems involving spatio-temporal evolution such as radiation ("in a distance of $x$ you witness the radiated information of the source which is now past at the source").

Mathematically you can see that as a transformation to a set of variables with more appropriate features (usually less complicated terms).

Let's end it with a source recommendation. For such questions is that definitely Hamilton's and Blackstock's Nonlinear Acoustics, specifically chapter 3, where the KZK equation is derived as a special case of the second order wave equation.

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