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Math Help - Group velocity of a wave

  1. #1
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    Group velocity of a wave

    The phase velocity for waves in a determined material is given by: v= v=C_1 + C_2 \lambda

    Both Cs are constants. \lambda is wavelength. Determine the group velocity v_g!

    My calculation:

    \omega=v \cdot k

    k=\frac{2 \pi}{\lambda}

    v_g=\frac{d \omega}{dk} = \frac{d(vk)}{dk}= (C_1 +C_2 \lambda)'k + (C_1 +C_2 \lambda)k' = C_2k + (C_1 + C_2 \lambda)(-\frac{2 \pi}{\lambda^{2}})


    This is unfortunately incorrect answer!

    This is physics, but purely a mathematical difficulty.
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  2. #2
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    Re: Group velocity of a wave

    Quote Originally Posted by techmath View Post
    The phase velocity for waves in a determined material is given by: v= v=C_1 + C_2 \lambda

    Both Cs are constants. \lambda is wavelength. Determine the group velocity v_g!

    My calculation:

    \omega=v \cdot k

    k=\frac{2 \pi}{\lambda}

    v_g=\frac{d \omega}{dk} = \frac{d(vk)}{dk}= (C_1 +C_2 \lambda)'k + (C_1 +C_2 \lambda)k' = C_2k + (C_1 + C_2 \lambda)(-\frac{2 \pi}{\lambda^{2}})


    This is unfortunately incorrect answer!

    This is physics, but purely a mathematical difficulty.
    \omega = v \cdot k = (C_1+C_2\lambda) \cdot \frac{2\pi}{\lambda} = \frac{2\pi \cdot C_1}{\lambda} + 2\pi \cdot C_2

    \frac{d\omega}{d\lambda} = -\frac{2\pi \cdot C_1}{\lambda^2}


    \lambda = \frac{2\pi}{k}

    \frac{d\lambda}{dk} = -\frac{2\pi}{k^2}


    note that ...

    \frac{d\omega}{dk} = \frac{d\omega}{d\lambda} \cdot \frac{d\lambda}{dk}
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  3. #3
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    Re: Group velocity of a wave

    Thank you very much.

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