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Wave equations (Video)

The wave equation

Let us recall the wave equation

$$ \left( \frac{1}{c^2}\frac{\partial^2 }{\partial t^2} - \frac{\partial^2 }{\partial x^2} \right) \phi = 0. $$

Any well-behaved function $\phi(x,t)=f(x-ct)$ or $f(x+ct)$ is a solution. In particular, the functions

$$ \phi(x,t) = \sin(kx-\omega t) $$

are solutions if $\omega=ck$ for any $k.$

[We have $\phi(x,t)=\sin[k(x-ct)$.]

A general solution is written as

$$ f(x-ct) = \int [a(k) \cos(kx-\omega t) + b(k) \sin(kx-\omega t)] dk. $$

We can call it a wave packet and it can have the form of a localised function.

Question. Describe the motion of the wave packet.