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Splitting wave equation into x-component and y-component

  1. Aug 2, 2010 #1
    Hello,

    I've been working for a while with the following wave equation PDE:

    [tex]
    \[
    \frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }} = A\frac{{\partial ^2 p}}{{\partial t^2 }} + B\frac{{\partial p}}{{\partial t}}
    \]
    [/tex]

    In preparation for the application of a Perfectly Matched Layer (PML), I need to be able to split the equation into x-components and y-components. This means that I would like to write down two equations. One equation is the x-projection of the equation above, and the other equation is the y-projection.

    In the PML literature, examples are shown of Ampere's Law being written in two equations. One equation is for the x-projection, and the other is for the y-projection. I would like to do the same for the equation above. Where do I begin?

    Then once I numerically solve the equation using a Finite-Difference Time-Domain method, I would like to combine the pressure components [tex]p_x [/tex] and [tex]p_y[/tex] to get [tex]p[/tex] again. How would I do this?
     
    Last edited: Aug 2, 2010
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  3. Aug 2, 2010 #2

    arildno

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    Not quite sure what you mean here.

    Are you talking about separation of variables?

    That will do nicely here, as long as A and B are, at most, functions of "t" only!
     
  4. Aug 2, 2010 #3
    Hello arildno,

    Thank you for your response!

    Yes, A and B are functions of "t" only. What I need to do is re-write the equation as two equations, one with [tex]p_x [/tex] and the other with [tex]p_y [/tex].
     
  5. Aug 2, 2010 #4
    Perhaps let me clarify: one equation is the x-projection of the original equation, and the other equation is the y-projection.
     
  6. Aug 2, 2010 #5

    arildno

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    Well, if P(x,y,t)can be written as
    [tex]P(x,y,t)=P_{x}(x)*P_{y}(y)*P_{t}(t)[/tex],
    then, by inserting into your diff.eq, and dividing with P, gives you:
    [tex]\frac{1}{p_{x}}p_{x}^{''}(x)+\frac{1}{p_{y}}p_{y}^{''}(y)=A\frac{1}{p_{t}}p_{t}^{''}(t)+B\frac{1}{p_{t}}p_{t}^{'}(t)[/tex]
    This gives rise to three basic linear equations, one for each variable, by identifying what set of terms must equal some constant.
     
  7. Aug 2, 2010 #6
    Hello arildno,

    Once again, thank you very much for your response; this is greatly appreciated! Let me see if I understand what you are saying.

    Now [tex]P(x,y,t)=P_x(x) P_y(y) P_t(t) [/tex], which is the product of each individual component. This is inserted into the wave equation PDE, and then both sides of the equation is divided by [tex]P(x,y,t)[/tex]. I understand this separation of variables approach.

    Now how do I obtain a system of three linear equations? Does this relate in any fashion to projections in vector calculus? Would one equation describe the projection in the x-direction and another in the y-direction?
     
  8. Aug 2, 2010 #7
    Another question: Why is [tex]P(x,y,t)=P_x(x) P_y(y) P_t(t)[/tex], and what is the physical reason for expressing this as a product of functions? It is mathematically feasible, but I am curious with respect to why it is physically reasonable. (It appears to be physically reasonable as well.)
     
  9. Aug 2, 2010 #8

    arildno

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    Hmm.. I am sorry.

    I am unacquainted with the perfectly matched layer technique (PML), which seems to be a coordinate transformation, taking advantage of the complex plane.

    I assume this is where your "x-projections/y-projections" comes into fore, I was unsure what you meant by those terms, given that you have a scalar function.

    Clearly, what you are asking about, is how to perform a smart coordinate transformation for your diff.eq.

    There, I cannot help you, I'm afraid.

    The technique of "separation of variables" is a general technique of limited applicative value, there are lots of diff.eqs, due to boundary&initial conditions that cannot be handled by this.

    I am quite certain that the PML-technique has been designed to tackle a number of such cases where simple techniques, like that of S.O.V. are not applicable.

    Sorry for having led you astray here.
     
  10. Aug 2, 2010 #9
    Hello arildno,

    Once again, thank you very much for your reply. No, I don't think that you've led me astray, and the advice that you give is quite good. Let me perhaps add more information to the problem.

    From what I know of the literature, there are many ways of applying a Perfectly Matched Layer (PML). I am actually interested in applying a Convolutional Perfectly Matched Layer (CPML), which is described here in this paper by Roden and Gedney:

    http://www.engr.uky.edu/~gedney/roden_gedney_cpml_motl.pdf [Broken]

    From reading the literature, what I understand is that the original equation must be written as two separate equations.

    The authors of this paper take the x-projection of Ampere's Law and then show how the approach can be applied only to the x-projection. (Such as on page 3 of this paper.)

    In a similar approach to taking the x-projection and y-projection of Ampere's Law, I would like to take the

    x-projection
    y-projection

    of this equation:

    [tex]
    \[
    \frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }} = A\frac{{\partial ^2 p}}{{\partial t^2 }} + B\frac{{\partial p}}{{\partial t}}
    \]
    [/tex]
     
    Last edited by a moderator: May 4, 2017
  11. Aug 2, 2010 #10
    Note that the LHS of my equation can be written using the Laplace operator. Perhaps some vector math could be applied here and the equation could be split into x-components and y-components?
     
  12. Aug 2, 2010 #11

    arildno

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    Thank you for that link!
    It has som vague resemblance to conformal mapping techniques, I think, that I was fleetingly introduced to many years ago.

    I will try and read that link and see if I'm up to it still.
     
  13. Aug 2, 2010 #12
    Sure, thanks arildno. I've worked with PML-type techniques in the past, but what puzzles me is how to take an equation (such as my equation) and then split it up into two equations.

    One equation is for the x-projection, and the other equation is for the y-projection. This reminds me of vector projections.

    Roden and Gedney simply give the reader the x-projection equation, but they don't say how the equation was derived from Ampere's Law. There is a similar equation for the y-direction as well, I assume, but since the focus of the paper is on the application of the PML, they only give an example.

    I think that I understand how to apply the PML, but the challenging part (at least for me) is how to split the equation into an x-projection and a y-projection.
     
  14. Aug 2, 2010 #13
    Hello arildno,

    Looking at this again, I think that you were completely correct with your initial suggestion and I didn't really understand what you were saying. My apologies.

    The substitution [tex]p = p_x p_y p_t [/tex] is simply a substitution which is mathematically reasonable. I believe that something similar to this substitution is being done here:

    http://www.mathpages.com/home/kmath242/kmath242.htm

    So what we are trying to do is to find three equations which correspond to components [tex] {p_x,p_y, p_t} [/tex]. Define coefficients [tex]{k_x,k_y,k_t}[/tex], and then it follows that:

    [tex]
    \[
    \frac{1}{{p_x }}\frac{{\partial ^2 p_x }}{{\partial x^2 }} + \frac{1}{{p_y }}\frac{{\partial ^2 p_y }}{{\partial y^2 }} = \frac{A}{{p_t }}\frac{{\partial ^2 p_t }}{{\partial t^2 }} + \frac{B}{{p_t }}\frac{{\partial p_t }}{{\partial t}}
    \]
    [/tex]

    [tex]
    \[
    k_x = \frac{1}{{p_x }}\frac{{\partial ^2 p_x }}{{\partial x^2 }}
    \]
    [/tex]


    [tex]
    \[
    k_y = \frac{1}{{p_y }}\frac{{\partial ^2 p_y }}{{\partial y^2 }}
    \]
    [/tex]

    [tex]
    \[
    k_t = \frac{A}{{p_t }}\frac{{\partial ^2 p_t }}{{\partial t^2 }} + \frac{B}{{p_t }}\frac{{\partial p_t }}{{\partial t}}
    \]
    [/tex]

    [tex]
    k_t = k_x + k_y
    [/tex]

    Does this look reasonable to you? Essentially what we have are three separate equations, each involving components [tex]{p_x, p_y, p_y} [/tex]. Now to get the pressure [tex] p [/tex] we simply calculate:

    [tex]p = p_x p_y p_t [/tex]

    If my understanding is correct, then I can easily proceed with the application of the convolutional PML as described in the paper.

    So once again, thank you very, very much for your help! :smile:
     
    Last edited: Aug 2, 2010
  15. Aug 2, 2010 #14
    Furthermore, for initial conditions, if [tex]p=0[/tex], then I think that it might also be possible to assume that [tex]k_x = 0[/tex], [tex]k_y=0[/tex], and [tex]k_t = 0[/tex].
     
  16. Aug 3, 2010 #15

    arildno

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    I'm not as sure as you.

    You see, separation of variables requires some very particular sets of initial&boundary conditions in order to work properly.

    It is not, therefore, a wholly general procedure, rather, it is a bonus we sometimes are granted.

    Intelligent co-ordinate transformations however, where for example, a free surface is taken care of in that transformation, so that we change our diff.eq into one that can be solved on a fixed domain instead, is wholly general.

    And from what I can glean, that is what PML is about.
     
  17. Aug 3, 2010 #16
    Hello arildno,

    Many thanks for your reply!

    Well, from what I understand, the separation of variables approach simply produces three equations. These equations are then subjected to the application of the Convolutional Perfectly Matched Layer (CPML). So I think that the separation of variables approach is different from the application of the CPML. As far as I know, the separation of variables simply gets the equations into a form where I can apply the CPML.

    So I think that to apply the CPML, there would be a convolution applied to the separated equations. The convolution is denoted by the * operator:

    [tex]
    \[
    k_x = \frac{1}{{p_x }}\frac{{\partial ^2 p_x }}{{\partial x^2 }}*s_x \left( t \right)
    \]
    [/tex]

    [tex]
    \[
    k_y = \frac{1}{{p_y }}\frac{{\partial ^2 p_y }}{{\partial y^2 }}*s_y \left( t \right)
    \]
    [/tex]

    Since I am solving these equations numerically using the Finite-Difference Time Domain (FDTD) method, I'll go ahead and discretize these PDEs and see if the solution works. I'll then post my findings here.

    Once again, thank you very much for your suggestions.
     
  18. Aug 3, 2010 #17
    I've started to numerically discretize the equations that I've listed in a previous post, but I've run into a difficulty.

    At each timestep, I need to solve for [tex] {p_x,p_y,p_t} [/tex], but at the same time I don't know [tex] {k_x,k_y, k_t} [/tex].

    Is there some sort of physical basis for calculating [tex] {k_x,k_y, k_t} [/tex]? What is the physical meaning of these three constants?
     
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