1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

PDE, 2D Laplace Equation, Sep. of Variables, Finding Potential

  1. Feb 19, 2012 #1
    1. The problem statement, all variables and given/known data
    A square rectangular pipe (sides of length a) runs parallel to the z-axis (from [itex]-\infty\rightarrow\infty[/itex]). The 4 sides are maintained with boundary conditions
    (i) V=0 at y=0 (bottom)
    (ii) V=0 at y=a (top)
    (iii) V=constant at x=a (right side)
    (iv) [tex]\frac{\partial V}{\partial x}=0[/tex] at x=0 (left side).
    Use separation of variables to find the potential V(x, y, z) inside the pipe.

    2. Relevant equations
    General solutions for 2D Laplace equation
    [tex] X(x)=Ae^{kx}+Be^{-kx}\qquad Y(y)=C\cos ky+D\sin ky[/tex] where A, B, C, and D are constants.

    3. The attempt at a solution
    I know since that C=0 in order to satisfy the boundary conditions V=0 for y=0 and y=a. I however do not know how to satisfy the boundary condition for the left side where dV/dx=0.
    This is what I've tried:
    [tex]\frac{d}{dx}[Ae^{kx}+Be^{-kx}]=kAe^{kx}-kBe^{-kx}[/tex]
    and set A=0 because [itex]kAe^{kx}[/itex] does not go to zero.
     
    Last edited by a moderator: Feb 19, 2012
  2. jcsd
  3. Feb 19, 2012 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Set the derivative equal to 0. This gives you a relationship between A and B.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: PDE, 2D Laplace Equation, Sep. of Variables, Finding Potential
Loading...