classical mechanics - Pulse with envelope propagation

A propagating plane wave can be written as

$$f(z,t) = A_0 \cos \left( \omega_0 t - k_0 z \right)$$

and it moves along the positive $z$ with velocity $v = \omega_0 / k_0$.

Let's now consider this gaussian function: $g(z = 0, t) = e^{-at^2}$. If it is assumed as the envelope of $f(z = 0, t)$, it will be

$$h(z = 0, t) = A_0 e^{-at^2} \cos \left( \omega_0 t \right)$$

Now, how can be $h$ be expressed for a generic $z$, in order to define a propagating cosine enveloped by the gaussian function, with both the cosine and the envelope moving at the same velocity $v$?

Answer

In a non-dispersive medium, yes. Here all waves move at the same phase velocity $c=\omega_0/k_0$, and the waveform is given by

$$h(z,t) = A_0 e^{-a(t-z/c)^2}\cos(\omega_0t-k_0z).$$ If your medium is dispersive this will change, and it will change in different ways depending on the nature of the dispersion.