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

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.