1. Not finding help here? Sign up for a free 30min 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!

Laplaces equation in polar coordinates

  1. Mar 20, 2007 #1
    The function [tex] u(r,\theta) [/tex]

    satisfies Laplace's equation in the wedge [tex] 0 \leq r \leq a, 0 \leq \theta \leq \beta [/tex]

    with boundary conditions [tex] u(r,0) = u(r,\beta) =0, u_r(a,\theta)=h(\theta) [/tex]. Show that

    [tex] u(r,\theta) = \sum_{n=0}^\infty A_nr^{n\pi/\beta}sin(\frac{n\pi\theta}{\beta}) [/tex]

    [tex]A_n=a^{1-\frac{n\pi}{\beta}\frac{2}{n\pi}\int_{0}^{\beta}h(\theta)sin\frac{n\pi\theta}{\beta}d\theta [/tex]
     
    Last edited: Mar 21, 2007
  2. jcsd
  3. Mar 21, 2007 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You have posted here before so surely you know the basic rules!

    No one is going to do your homework for you and it wouldn't help you if they did! Show us what you have tried so we can see where you went wrong or got stuck.
     
  4. Mar 21, 2007 #3
    I saw the question, realised I was incapable and thought I'd put it up for some hints... I do have a similar problem viz laplace in the unit circle.....


    [tex] \nabla^2U=0 [/tex]

    Boundary conditions are 1) U=0 at r=0

    2) [tex] U(1,\theta)=2cos\theta [/tex]

    now I have quoted from the notes that the general solution is

    [tex] U(r,\theta) =C_0lnr + D_0 + \sum_{0}^\infty(C_0r+\frac{D_0}{r^n}).(A_ncosn\theta + B_nsinn\theta) [/tex]

    now I am told that B.C 1 implies [tex] D_0 = 0=C_n [/tex] for n=0,1,2,3,4....

    I am immediately confused why it is necessary to have these two coeffiecient set to zero, surely we could have some situation whereby the three terms could cancel to zero without insisting their coefficients are zero??
     
    Last edited: Mar 21, 2007
  5. Mar 21, 2007 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    The sines, cosines and the constant function are linearly independent on the unit circle.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Laplaces equation in polar coordinates
Loading...