# How to solve general wave equation and dispersion relation using Fourier series? [closed]

In this paper (open access), the authors used Fourier series with most general wave equation to find the dispersion relation. I am presenting some main equations as snippets to depict their solution. We want to solve for $$u(x,t)$$ in the following wave equation (Eq. (1) in the paper):

$$$$\tag{1} \frac{\partial}{\partial x}\left[ E(x,t)\frac{\partial}{\partial x} u(x,t) \right]-\frac{\partial}{\partial t}\left[ \rho(x,t)\frac{\partial}{\partial t} u(x,t) \right]=0.$$$$

So we expand or represent the quantities using Fourier series:

$$\tag{2} E(x,t)=\sum_{p=-\infty}^\infty E_{p}e^{ip(\omega_mt-k_mx)},$$ and $$\tag{3} \rho(x,t)=\sum_{p=-\infty}^\infty \rho_{p}e^{ip(\omega_mt-k_mx)},$$ both having periodicty of $$\omega_m$$ and $$k_m$$ for time and space. Now we assume a general solution (Floquet form) for $$u(x,t)$$ as:

$$\tag{4} u(x,t)=e^{i(\omega t-kx)} \sum_{n=-\infty}^\infty u_{n}e^{in(\omega_mt-k_mx)},$$

and plug everything in the wave-equation. The authors say that using orthogonality we directly have the equation (9 in the paper):

$$\tag{5} \sum_{n=-\infty}^\infty [(k+nk_m)(k+pk_m)]E_{p-n}u_n = \sum_{n=-\infty}^\infty [(\omega+n\omega_m)(\omega+p\omega_m)]\rho_{p-n}u_n.$$

Then they say that it can be "recast" in the form (Eq. 11 in paper):

$$\tag{6} (\omega^2 \hat{L}_2 + \omega \hat{L}_1 + \hat{L}_0)\vec{u}=0,$$ which is a matrix equation in $$\vec{u}$$ and connects $$u_n$$, $$u_{n−1}$$, and $$u_{n+1}$$.

How can Eq. 5 be cast in Eq. 6 given that Eq. 5 does not have a relation between $$u_n$$, $$u_{n−1}$$, and $$u_{n+1}$$? It only has two independent sums on both sides. They do not have different phases on both side to shift a phase term and take common summation to relate these.

• Siddhant, are you still working on this? Oct 19, 2021 at 20:48
• This question has been closed as it seems to be abandoned. It can be reopened if someone wants to add an answer or the OP addresses questions/suggestions in the comments.
– Tyberius
Jan 11 at 20:37