# Asymptotic disperson of 2D wave equation with 1D periodic potential

+ 1 like - 0 dislike
102 views

Consider the following 2D wave equation:
$$\left(\frac{d^2}{dx^2}-k_y^2+\omega^2\ V(x)\right)\psi(x)=0$$
where $V(x+L)=V(x)>0$ is a positive periodic potential, $k_y$ is the wave vector along $y$-direction, $\omega$ is the frequency, and the eigenfunction satisfies the periodic boundary condition $\psi(x+L)=\psi(x)$, $\frac{d}{dx}\psi(x+L)=\frac{d}{dx}\psi(x)$.

From the numerical solutions of several different periodic potentials $V(x)$, I find that all bands approach to linear asymptotic dispersion $\omega_n(k_y)$ as $k_y\rightarrow\infty$. And it seems that the asymptotic group velocities for different bands are identical and are only determined by the maximum of the potential $V(x)$, namely

$$\lim_{k_y\rightarrow\infty}\frac{d\omega_n}{dk_y}=1/\sqrt{V_{\mathrm{max}}}.$$

The following figures show two examples (the first 9 bands in each case). However, I cannot prove this conjecture. Can some one help me prove or disprove this conjecture?

The insets in the upper subfigures are the profiles of the potential function $V(x)$ in one period.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:$\varnothing\hbar$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.