Can anyone identify this series trick?

[Deiniol]

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 σ_r 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.

 

[Erland]

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.

 

[Deiniol]

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=σ_rt+Φ₁-Φ₁' where σ_r=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.