Wave pulse

A wave pulse has a finite extent in space and time. Some examples are,

$ y= A\exp(-(kx-\omega t+\varphi)^2)$
$ y= A\exp(-\alpha(kx-\omega t+\varphi)^2)\cos(kx-\omega t+\varphi)$
$ y= A(\text{H}(2+kx-\omega t+\varphi)-\text{H}(kx-\omega t+\varphi))$

Here $\text{H}(x)$ is the Heaviside function.

$y$

$x$

A = 1 [m]

α = 0.01 

k = 2 [rad/m]

ω = 1 [rad/s]

φ = 0 [rad]

$\lambda=\frac{2\pi}{|k|} = $

$T=\frac{2\pi}{|\omega|} = $

$c = \frac{\omega}{k} = $ [m/s]

$t=$

If $k\omega >0$ the wave moves in the $+x$ direction and if $k\omega <0$ the wave moves in the $-x$ direction. The wave speed is $c=\frac{\lambda}{T}=\lambda f=\frac{\omega}{k}$.

All of the example functions can be written as a function of a single variable $\xi$.

$ y= A\exp(-\xi^2)$
$ y= A\exp(-\alpha\xi^2)\cos(\xi)$
$ y= A(\text{H}(2+\xi)-\text{H}(\xi))$

where $\xi=kx-\omega t+\varphi$. Any function of a single variable can be used to make a wave pulse.

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