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Proving general solution of Helmholtz equation

  1. Oct 30, 2013 #1
    1. The problem statement, all variables and given/known data

    Prove that F(k•r -ωt) is a solution of the Helmholtz equation, provided that ω/k = 1/(µε)1/2, where k = (kx, ky, kz) is the wave-vector and r is the position vector. In F(k•r -ωt), “k•r –ωt” is the argument and F is any vector function.

    2. Relevant equations
    Helmholtz Equation: ∇2A+k2A=0


    3. The attempt at a solution
    I first completed the dot of k and r thus,
    F(xkx+ yky+zkz)


    I am a little confused at this point, if I plug it into the equation I don't see how it would be 0
     
    Last edited: Oct 30, 2013
  2. jcsd
  3. Oct 30, 2013 #2

    SteamKing

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    It's not clear what you've done, since you haven't shown your work.
     
  4. Oct 30, 2013 #3
    I completed the dot between k and r within F.

    k = (kx, ky, kz)
    TA said to use r=(x,y,z)

    thus the function becomes
    F(xkx+ yky+zkz)

    2F(xkx+ yky+zkz)=0, right?


    but then how would k2F(xkx+ yky+zkz)=0?
     
  5. Oct 30, 2013 #4

    BruceW

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    2F(xkx+ yky+zkz)=0 is not right. Because then, as you say, you would need k2F=0, which is the trivial solution.

    It is a kind of weird problem in the first place, since the Helmholtz equation does not care about time dependence. Are you sure the problem wasn't to show that ##F(k\cdot r -\omega t)## is a solution of the wave equation, and to go on to prove that it is also a solution of the Helmholtz equation?
     
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