# Derivative with respect to a difference of independent variables

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)

• 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. Apr 22, 2016 at 19:33
• @PeterDiehr do you know, can this equation be transformed to a usual PDE (without delay) like the Burger's one? Apr 23, 2016 at 21:50
• I don't think so; that's why it is always solved numerically, according to the article. Apr 23, 2016 at 23:33

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").