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Prove a satisfaction with the wave equation

  1. Jan 7, 2016 #1
    1. The problem statement, all variables and given/known data
    i want to prove that the functions u(r,t)=(1/r)f(r-v*t) and u(r,t)=(1/r)f(r+v*t) satisfy the wave equation in spherical coordinates, i have tried a lot to solve it but in each time i would face a problem.

    2. Relevant equations
    wave equation : grad^2(u)=(1/v)*(partial ^2 u/partial t ^2)

    3. The attempt at a solution i have tried to solve it in different ways but it does not work with me.
     
  2. jcsd
  3. Jan 7, 2016 #2
    it is v^2 not v
     
  4. Jan 8, 2016 #3

    blue_leaf77

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    What did you get after evaluating ##\nabla^2u##?
    Since ##u## is not a function of ##\theta## and ##\phi##, the Laplacian operator will look like
    $$
    \frac{1}{r^2}\frac{\partial}{\partial r} \left( r^2 \frac{\partial u}{\partial r} \right)
    $$
     
    Last edited: Jan 8, 2016
  5. Jan 8, 2016 #4
    i am rely sorry ,i am a new member here and i don't know how to write the symbols correctly ...((i found grad ^2 by saying that it equals the second partial derivative for u with respect to r)) .and i found the second partial derivative for u with respect to time. finally i could not match between them , usually when i evaluate the last i would find the first in it and so we can substitute here and reach to our goal,but it didn't work this time ,the question wants a general solution .a have solved such a question when u=sin(x-v*t) it was easy .
    thank you so much ^-^
     
  6. Jan 8, 2016 #5

    blue_leaf77

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    Your wave equation is correct, taking into account the correction you gave in post #2. If you can't match the left and right hand side then you must be doing something wrong. Since you said that you have calculated ##\partial^2 u/\partial t^2##, can you show what you got here?
     
  7. Jan 8, 2016 #6
    it goes like :
    ∂u/∂t=(1/r)(∂f/∂t)+(0)*f
    =(1/r)(∂f/∂t)(-v)
    ∂²u/∂t²=(v²/r)*(∂²f/∂t²)
     
  8. Jan 8, 2016 #7

    blue_leaf77

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    That doesn't seem to be quite correct, especially the way you arrived at the second line. From the first line you have ##\frac{1}{r}\frac{\partial f}{\partial t}##. To do the partial derivative w.r.t ##t##, you would have to use the chain rule, upon which it will be
    $$
    \frac{1}{r}\frac{\partial f}{\partial t} = \frac{1}{r}\frac{\partial f}{\partial (r-vt)} \frac{\partial (r-vt)}{\partial t} = \frac{-v}{r}f'(r-vt)
    $$
    where ##f'(r-vt) = \frac{\partial f}{\partial (r-vt)}##. Calculating ##\partial^2 u/\partial t^2##, what did you get?
     
    Last edited: Jan 8, 2016
  9. Jan 8, 2016 #8
    i went through the first solution that you gave me,Since u is not a function of θ and ϕ as you have said ,and it is going real good ,i guess this is the best way ,i got two equation and i just have to substitute one in the other..thank you so much bro.
     
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