|Feb11-13, 08:19 AM||#1|
Simple PDE Question
It's been a little too long since I've has to do this. Can someone please remind me, how do you get from:
∂u/∂t = C(∂u/∂g)
∂^2u/∂t^2 = (C^2)(∂^2u/∂t^2)
The notation here is a little clumsy, but I'm just taking the second PDE of each side. How does the C^2 get there? Seems like it ought to be C but I can't put my finger on a proof either way.
By the way, this comes up in a derivation of the wave equation:
∂^2u/∂x^2 = (1/c^2)(∂^2u/∂t^2)
u(x,t) = u(x ± ct)
I'm sure someone out there knows this. Thanks for your help.
|Feb11-13, 02:42 PM||#2|
|Feb11-13, 03:43 PM||#3|
LCKurtz, thanks for the response. Alright, here goes.
Starting from a general function u(x - ct), define g=x - ct. 
So we have ∂u/∂x = (∂u/∂g)(∂g/∂x) and ∂u/∂t = (∂u/∂g)(∂g/∂t) . 
The PDEs from  are: ∂g/∂x = 1, and ∂g/∂t = - c . 
So from  and , ∂u/∂x = ∂u/∂g . 
The second PDE from  is ∂2u/∂x2 = ∂2u/∂g2, is that correct? 
Also from  and , ∂u/∂t = -c(∂u/∂g) . 
Now, to get from  and  to the wave equation ∂2u/∂x2 = (1/c2)(∂2u/∂t2)
seems to require, from , ∂2u/∂t2 = (c2)(∂2u/∂g2)
It's that last step I don't quite get, unless - which is by no means unlikely - I'm making an error someplace else. Seems like the c2 should just be c .
The context here is I'm an electrical engineer trying to understand the physics or ultrasound transmission through a waveguide. This derivation comes from "Basics of Biomedical Ultrasound for Engineers", Axhari, 2010.
|Feb11-13, 08:03 PM||#4|
Simple PDE Question
|Feb13-13, 07:49 AM||#5|
Okay, I get it now. I needed to carry out the second PDEs one more step and "chain rule" it. Thanks for your help.
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