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!

Associated Legendre polynomial (I think)

  1. Oct 21, 2015 #1
    1. The problem statement, all variables and given/known data
    I'm not 100% sure what this type of problem is called, we weren't really told, so I'm having trouble looking it up. I'd really appreciate any resources that show solved examples, or how to find some!

    Anyway. For the solution to the spherical wave equation φ(t, θ, Φ)
    i) Calculate φ( 0, θ, Φ) and ##\frac{\partial φ}{\partial t}##(0, θ, Φ) in terms of Anm and Bnm.

    ii)Let the initial value of φ be given by φ( 0, θ, Φ) = ##Y_{3}^{-2}##
    and
    ##\frac{\partial φ}{\partial t}##(0, θ, Φ) = 16##Y_{2}^{2}##

    2. Relevant equations
    ##P_{n}^{m}(z) = \frac{1}{2^n n!}(1-z^2)^{0.5m} \frac{d^(n+m)}{dz^(n+m)}(z^2-1)^n## (1)

    ##Y_{n}^{m}(\theta, \phi) = (\frac{2n+1}{4 \pi} \frac{(n-m)!}{(n+m)!})^{0.5} e^{im \phi}P_{n}^{m}(cosθ)## (2)

    General solution:
    φ = ΣΣ ##A_{nm} cos(t \sqrt{n(n+1)}) + B_{nm}sin(t \sqrt{n(n+1)})##

    3. The attempt at a solution

    I've done part one.
    φ( 0, θ, Φ) = ΣΣ ##A_{nm}Y_{n}^{m}(\theta, \phi)## from just subbing in t= 0 to the general solution.

    ##\frac{\partial \phi}{\partial t}(0,\theta,\phi) =##ΣΣ##B_{nm} \sqrt{n(n+1)}Y_{n}^{m}(\theta, \phi)##

    But for part two, I don't know what to do with the initial values! If φ( 0, θ, Φ) = ##Y_{3}^{-2}## then does that mean m = -2 and n = 3 for the whole equation? Presumably not or ##A_{nm}## = 0.

    And I can't actually evaluate ##Y_{3}^{-2}##. Because subbing m and n into equation two means I have to also work out P from equation 1. So sub in z = cosθ in this case, but how can I differentiate with respect to cosθ? That doesn't make sense!
     
    Last edited: Oct 21, 2015
  2. jcsd
  3. Oct 21, 2015 #2
    OK, worked out how to evaluate ##Y_{3}^{-2}##, and got it equals ##\sqrt{210} e^{(-2i \theta)} \frac{cos(\theta)sin(\theta)^2}{8}##, which is the equal to
    ∑∑ ##A_{nm}Y_{n}^{m}##
    Where the first sum is from n=0 to infinity and the second is from m=-n to n. So I don't know how to find ##A_{nm}## from that?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Associated Legendre polynomial (I think)
  1. Legendre Polynomials. (Replies: 0)

  2. Legendre polynomials (Replies: 1)

  3. Legendre polynomials (Replies: 4)

Loading...