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Math Help - wave equation

  1. #1
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    wave equation

    I have already shown T(x,t) = (A/sqrt(t))*exp((-x^2)/(4kt))

    is a solution of the wave equation

    I now need to show int (T(x,t)) dx between infinity and neg infinity is a constant function of T

    obvioulsy I cant integrate the function.. I know integrating the wave equation would give me [(t^2)/2] is this enough to say?
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  2. #2
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    Quote Originally Posted by James0502 View Post
    I have already shown T(x,t) = (A/sqrt(t))*exp((-x^2)/(4kt))

    is a solution of the wave equation

    I now need to show int (T(x,t)) dx between infinity and neg infinity is a constant function of T

    obvioulsy I cant integrate the function.. Mr F says: Yes you can. See below.

    I know integrating the wave equation would give me [(t^2)/2] is this enough to say?
    To find I = \frac{A}{\sqrt{t}} \int_{-\infty}^{+ \infty} e^{-\frac{x^2}{4 kt}} \, dx I suggest making the substitution u = \frac{x}{2 \sqrt{kt}}.

    Note that \int_{-\infty}^{+ \infty} e^{-u^2} \, du is a famous definite integral. See Gaussian Integral -- from Wolfram MathWorld
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  3. #3
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    Brilliant, thankyou

    I have another wave equation problem

    http://people.maths.ox.ac.uk/~earl/sheet5b.pdf

    question 3

    I have verified it is a solution

    now applying ICs I get

    sum {sin(n.pi.x/l)[A_n] = alpha sin(pi.x/l)}

    which I cant see how I would get A_n

    many thanks
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by James0502 View Post
    Brilliant, thankyou

    I have another wave equation problem

    http://people.maths.ox.ac.uk/~earl/sheet5b.pdf

    question 3

    I have verified it is a solution

    now applying ICs I get

    sum {sin(n.pi.x/l)[A_n] = alpha sin(pi.x/l)}

    which I cant see how I would get A_n

    many thanks
    This may help you.
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  5. #5
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    Quote Originally Posted by James0502 View Post
    I have already shown T(x,t) = (A/sqrt(t))*exp((-x^2)/(4kt))

    is a solution of the wave equation

    I now need to show int (T(x,t)) dx between infinity and neg infinity is a constant function of T

    obvioulsy I cant integrate the function.. I know integrating the wave equation would give me [(t^2)/2] is this enough to say?
    You say that the given solution is a solution of the wave equation. What equation do you refer to? When I think wave equation, I think

    T_{tt} = c^2 T_{xx}

    (although I've seen solutions like this of the heat equation T_t = k T_{xx}).
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  6. #6
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    erm.. I have dT/dt = k.dT^2/Dx^2
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  7. #7
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    Quote Originally Posted by James0502 View Post
    erm.. I have dT/dt = k.dT^2/Dx^2
    That's the heat equation.
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  8. #8
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    oops.. sorry - I meant heat equation!
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