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

  • Thread starter trncell
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Homework Statement


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.

Homework 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.

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.
 
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Answers and Replies

  • #2
vela
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Set the derivative equal to 0. This gives you a relationship between A and B.
 

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