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Simple PDE Questionby KenBakerMN
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#1
Feb1113, 08:19 AM

P: 11

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) to ∂^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) starting from u(x,t) = u(x ± ct) I'm sure someone out there knows this. Thanks for your help. 


#2
Feb1113, 02:42 PM

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PF Gold
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#3
Feb1113, 03:43 PM

P: 11

LCKurtz, thanks for the response. Alright, here goes.
Starting from a general function u(x  ct), define g=x  ct. [1] So we have ∂u/∂x = (∂u/∂g)(∂g/∂x) and ∂u/∂t = (∂u/∂g)(∂g/∂t) . [2] The PDEs from [1] are: ∂g/∂x = 1, and ∂g/∂t =  c . [3] So from [2] and [3], ∂u/∂x = ∂u/∂g . [4] The second PDE from [4] is ∂^{2}u/∂x^{2} = ∂^{2}u/∂g^{2}, is that correct? [5] Also from [2] and [3], ∂u/∂t = c(∂u/∂g) . [6] Now, to get from [5] and [6] to the wave equation ∂^{2}u/∂x^{2} = (1/c^{2})(∂^{2}u/∂t^{2}) seems to require, from [6], ∂^{2}u/∂t^{2} = (c^{2})(∂^{2}u/∂g^{2}) 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 c^{2} 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. 


#4
Feb1113, 08:03 PM

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PF Gold
P: 7,659

Simple PDE Question



#5
Feb1313, 07:49 AM

P: 11

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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