Def Dispersion
Thrm For a wave with general dispersion that
Missing \begin{aligned} or extra \end{aligned} f(x,t)={\frac{1}{2\pi}}\int_{-\infty}^{\infty}F(k,0)e^{-i\omega(k)t}e^{i k x}\,d k \\ f(x,t)={\frac{1}{2\pi}}\int_{-\infty}^{\infty}\!F(0,\omega)e^{-i k(\omega)x}e^{i\omega t}\,d\omega \end{aligned}$$ **Proof** Take a wave shaped in space, $f(x,0)$ with Fourier transform $F(k,0)$. Then by definition of Fourier transform, we have $$ f(x,0) = {\frac{1}{2\pi}}\int_{-\infty}^{\infty}F(k,0)e^{i k x}\,d k $$ To find the shape of the wave in space at a time $t$, rotate each $k$ with the right phase, which depends on $\omega$: $$ f(x,t)={\frac{1}{2\pi}}\int_{-\infty}^{\infty}F(k,0)e^{-i\omega t}e^{i k x}\,d k $$ Since $\omega$ is dependent on $k$ by dispersion, we have: $$ f(x,t)={\frac{1}{2\pi}}\int_{-\infty}^{\infty}F(k,0)e^{-i\omega(k) t}e^{i k x}\,d k $$ Similarly, we can get the second formula. **Def** <i><u>Narrow-Band Pulses</u></i> The narrow-band pulses are the pulses that the period is much smaller than the pulse width. ## Group and Phase Velocity **Def** <i><u>Group Velocity</u></i> For a pulse with $w$ and $k$ dependent, the group velocity is defined as $v_{g}={\frac{\mathrm{d}\omega}{\mathrm{d}k}}\bigg|_{k_c}$, which indicates the velocity of envelope. ^2f2e58 **Def** <i><u>Phase Velocity</u></i> The velocity of the ripples under the envelope are given by $v_{ph}=\frac{w_c}{k_c}$. We call this the phase velocity. **Prop** For any pulse $f(x,t)$ in the time and spacial space, we have: $$ f(x,t)={\frac{1}{2\pi}}e^{i k_{c}(v_{g}-\ v_{p h})t}\int_{-\infty}^{\infty}\!F(k)e^{i k(x-v_{g}t)}\,d k $$ >[!remark] > >The pulse $f(x,t)$ should be a real function. Actually the Fourier transform of the negative frequencies is missing in our formula. The correct answer should be added by the complex conjugate. [Symmetries](Fourier%20Transform.md#^ecbc8f)