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Separation of variables PDE

  1. Oct 21, 2015 #1
    1. The problem statement, all variables and given/known data
    The wave equation for ψ=ψ(t,x,y) is given by

    ##\frac{\partial ^2 \phi}{\partial t^2} - \frac{\partial ^2 \phi}{\partial x^2} - \frac{\partial ^2 \phi}{\partial y^2}##

    Use separation of variables to separate the equation into 3 ODEs for T, X and Y. Use the separation constants
    ##-k_{x}^{2}X## and ##-k_{y}^{2}Y##

    Do not introduce any more separation constants for T.
    2. Relevant equations


    3. The attempt at a solution
    I'm fairly sure I know how to start.
    Ansatz ψ(t,x,y) = T(t)X(x)Y(y). Sub the derivatives of this into the the wave equation:

    ##XY\frac{\partial ^2 T}{\partial t^2} - TY\frac{\partial ^2 X}{\partial x^2} - TX\frac{\partial ^2 Y}{\partial y^2}##=0

    Then divide by TXY:
    ##\frac{1}{T}\frac{\partial ^2 T}{\partial t^2} - \frac{1}{X}\frac{\partial ^2 X}{\partial x^2} - \frac{1}{Y}\frac{\partial ^2 Y}{\partial y^2}##=0

    Rearrange:
    ##\frac{1}{T}\frac{\partial ^2 T}{\partial t^2} = \frac{1}{X}\frac{\partial ^2 X}{\partial x^2} + \frac{1}{Y}\frac{\partial ^2 Y}{\partial y^2}##
    Which is only possible if LHS = RHS = constant, I think? But then I would get
    ##\frac{1}{X}\frac{\partial ^2 X}{\partial x^2} + \frac{1}{Y}\frac{\partial ^2 Y}{\partial y^2}## = constant, and how do I separate that further? Why would I need more than one constant of separation?
    Because if
    ##\frac{1}{X}\frac{\partial ^2 X}{\partial x^2} + \frac{1}{Y}\frac{\partial ^2 Y}{\partial y^2}## = constant

    Then surely each individual term must also be a constant, and I can just write
    ##\frac{1}{X}\frac{\partial ^2 X}{\partial x^2}## = ##-k_{x}^{2}X##
    ##\frac{1}{Y}\frac{\partial ^2 Y}{\partial y^2}## = ##-k_{y}^{2}X##

    And why wouldn't I introduce a new constant for T? What do I write instead,

    ##\frac{1}{T}\frac{\partial ^2 T}{\partial t^2}## = constant?
     
  2. jcsd
  3. Oct 21, 2015 #2

    pasmith

    User Avatar
    Homework Helper

    You are missing the "= 0" which would turn that expression into an equation.

    You mean
    [tex]
    \frac 1X \frac{\partial^2 X}{\partial x^2} = -k_x^2 \\
    \frac 1Y \frac{\partial^2 Y}{\partial x^2} = -k_y^2
    [/tex]

    Because you need [tex]
    \frac{1}{T}\frac{\partial ^2 T}{\partial t^2} - \frac{1}{X}\frac{\partial ^2 X}{\partial x^2} - \frac{1}{Y}\frac{\partial ^2 Y}{\partial y^2}
    = \frac{1}{T}\frac{\partial ^2 T}{\partial t^2} + k_x^2 + k_y^2 =0[/tex]
     
  4. Oct 21, 2015 #3
    Oh yes, I did mean that. An yep, get the bit about not introducing a new constant too! Thank you :)
     
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