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Question: I want t reduce the following wave equation $$u_n=c^2(u_{xx}+u_{yy}+u_{zz})$$ to Laplace equation $$u_{xx}+u_{yy}+u_{zz}+u_{\tau\tau}=0$$ by letting $\tau=ict$ and $i$ is imaginary. And I want to obtain the solution of wave equation in cylindrical coordinates via the solution of the Laplace equation. Assuming that $u(r,\theta,z,\tau)$ is independent of $z$.

Approach: I took $c^2={\tau^2}/{-t^2}$ and plugged it into this weird wave equation $u_n$. When I took the derivative $u_{\tau\tau}$ and tried finding the Laplace equation. I got it but for $t=\sqrt2$. After that I'm not sure how I am I supposed to solve this using the cylindrical coordinates for wave equation or Laplace equation. I would be grateful if anyone can atleast give me a direction! thank you

P.S The question was posed as it is in the Linear PDE by Myint-U and Debnath.

d13
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Assuming it's a typo there, and it's actually $$ u_{tt} = c^2 \left( u_{xx} + u_{yy} + u_{zz}\right ) $$ one can do the following (as mentioned in your textbook) $\tau = i ct$, so \begin{align} u_t &= u_{\tau} \tau_t = ic u_\tau \\ u_{tt} &= \left( u_t\right )_t = \left(ic u_\tau \right )_\tau \tau_t = i^2 c^2 u_{\tau \tau} = -c^2 u_{\tau \tau} \end{align} Substitute it to the wave equation $$ -c^2 u_{\tau \tau} = c^2 \left( u_{xx} + u_{yy} + u_{zz}\right ) \implies u_{\tau \tau} + u_{xx} + u_{yy} + u_{zz} = \Delta u= 0 $$ which is Laplace equation in $4$ dimensions $x,y,z,\tau$.

Kaster
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  • But this laplace I have to solve in 3D right?? Because u is independent of z?? – d13 Oct 23 '13 at 19:06
  • @d13, nope it's fully 4 dimensional equation now. It might not depend on $z$, but it should be some additional constraint, which is not there as it is right now. – Kaster Oct 23 '13 at 19:12
  • how am I supposed to solve it using cylindrical coordinates?? especially when it has $\tau$ in it. – d13 Nov 04 '13 at 16:26
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    @d13 You may look to this paper. But if in short, method is the same – separation of variables. – Kaster Nov 04 '13 at 19:12
  • thankyou Sir!! I appreciate your help and advice!!! you made my day. :)) – d13 Nov 05 '13 at 08:40