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Can anyone identify this series trick?

  1. Dec 15, 2015 #1
    Hi there, I am reading through a thesis and the author takes the infinite series:

    \begin{equation}
    u(x,t)=u_0(x)+u_1(x)\cos(\sigma t - \phi_1(x)) + u_1'(x)\cos(\sigma' t - \phi'_1(x))+\ldots
    \end{equation}

    and by letting σr be the difference between the frequencies σ and σ' changes the above to:

    \begin{equation}
    u=u_0+u_1\cos(n\sigma_rt-\phi_1) + u_1'\cos([n+\beta]\sigma_rt-\phi_1')+\ldots
    \end{equation}

    where β=±1 and n is an integer. I don't follow the manipulation from the first equation to the other, I just wondered if anyone is familiar with this trick and might be able to talk me through it. Many thanks.
     
  2. jcsd
  3. Dec 16, 2015 #2

    Erland

    User Avatar
    Science Advisor

    This would be unproblematic is if ##n=\sigma/\sigma_r## is not assumed to be an integer. It seems wrong otherwise, if ##u_1'=0## for example. So either this is wrong or ##n=\sigma/\sigma_r## should be assumed to be an integer, by some reason.
     
  4. Dec 17, 2015 #3
    Thanks for the reply Erland. The reason for the change is to eventually get u(x,t)|u(x,t)| into a form suitable for a Fourier expansion. I think that by letting θ=σt-Φ1 and α=σrt+Φ11' where σr=σ-σ' to transform (1) in the OP to:

    \begin{equation}
    u(x,t)=u_0+u_1\cos(\theta)+u_1'\cos(\theta-\alpha)+u_2\cos(2\theta-[\phi_2-2\phi_1])+u_2'\cos(2\theta-2\alpha-[2\phi_1'+\phi_2'-4\phi_1])+\ldots
    \end{equation}

    should work fine. Then u|u| can be expanded into a Fourier series of θ.
     
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