Wave packets

💼 Case: $1$D wave

In $1$D we have a forward travelling wave defined by $\phi(x,t)=e^{-i(kx-\omega t)}$ and this satisfies the $1$D wave equation

$$ \frac{\partial^2 \phi}{\partial x^2}=\frac{1}{c^2}\frac{\partial ^2\phi}{\partial t^2} $$

If we substitute in the travelling wave we find

$$ -k^2=\frac{1}{c^2}(-\omega^2) $$

or

$$ \omega=ck $$

🗒️ Note:

• A similar behaviour occurs for the imaginary part
• In general, one can have a forward wave ($+x$) and a backward travelling wave ($-x$) and the latter is given by $\phi=e^{i(kx+\omega t)}$

💼 Case: plane wave

• For a plane wave we use the $3$D wave equation $\phi(\vec x,t)=e^{i(\vec k \cdot \vec x-\omega t)}$

  $$ \nabla^2\phi=\frac{1}{c^2}\frac{\partial ^2\phi}{\partial t^2} $$

  Here we have $\omega /|k|=c$ a plane wave travelling along $\vec k$

  We need to recognise that $\vec k\cdot \vec x=\alpha$ is the equation of a plane perpendicular to $\vec k$ and whose distance from the origin is $\alpha/|\vec k|$.

  

🌴 Break: Lets go back to the $1$D we can sum these traveling waves along the $+x$ direction

$$ \phi(x,t)=\frac{1}{\sqrt{2\pi}}\int^\infin_{-\infin}G(k)e^{ik(x-ct)}\,\text dk $$

Here $G(k)$ is the Fourier transform of $\phi(x,0)$.

🗒️ Notes:

• Since all of the component waves travel at the same speed $c$ the shape of the wave is unchanged (it simply moves from left to right)
• This wave satisfies the wave equation

The wave equation is also applicable to non-dispersive media ( in which the phase velocity of the waves depend on the wavelength)

$$ v_p(k)=\frac{\omega(k)}{k} $$

<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/f322bbbc-7bed-4326-b2d1-1e0b2ffbe11b/general_wave.png" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/f322bbbc-7bed-4326-b2d1-1e0b2ffbe11b/general_wave.png" width="40px" /> A general wave (including dispersion) can be written as

$$ \phi(x,t)=\frac{1}{\sqrt{2\pi}}\int^\infin_{-\infin}G(k)e^{i(kx-\omega(k)t)}\, \text dk $$

where $\omega(k)$ is determined once we have solved the wave equation for $\phi$

🗒️ Note: since the components of the waves travel at different speeds, the wave will disperse and change shape during travel

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