Solving Chain Rule Problem with Equation (7.8)

[Robin04]

Homework Statement

https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=1007&context=foundation_wave
I'm trying to understand this paper and I'm stuck at equation (7.8). That part of the text is very short so I hope you don't mind if I don't copy the equations here.

Homework Equations

The Attempt at a Solution

I understand how I get (7.7). But if I want to take the second derivate I can just apply the chainrule again. $\partial_t^2{q} = \partial_t{[v(\partial_u{\tilde{q}}-\partial_s{\tilde{q}})]} = v[\partial^2_u{\tilde{q}}\partial_t{u}-\partial^2_s{\tilde{q}}\partial_t{s}] = v^2[\partial^2_u{\tilde{q}}+\partial^2_s{\tilde{q}}]$
But in the text it looks like if the author raised it to the power two (and got a different result) and I don't see why is that the same operation as taking the second derivative.

 

[Robin04]

I posted too soon, I realized what I did wrong. $\partial_u{\tilde{q}}$ is also dependent on s and so does the derivative with respect to s depend on u. With this, I got the same result.

 

[Geo_Zegarra2018]

First of all, what is your background in math? Do you understand what is second partial derivative?