Can anyone identify this series trick?

1. Dec 15, 2015

Deiniol

Hi there, I am reading through a thesis and the author takes the infinite series:

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

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

u=u_0+u_1\cos(n\sigma_rt-\phi_1) + u_1'\cos([n+\beta]\sigma_rt-\phi_1')+\ldots

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. Dec 16, 2015

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.

3. Dec 17, 2015

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=σrt+Φ11' where σr=σ-σ' to transform (1) in the OP to:

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

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