Expression of a Gaussian pulse in time domain

Hi,

I use Gaussian pulse for excited a thin wire in order to calculate the current at the center of the wire.

The expression of the Gaussian pulse is:

The direction of propagation is the vector Z
r is a position vector relative to the origin,
This Gaussian is linearly polarized along the vector X
Can you give me the exact expression of the scalar product r.k and the expression of this gaussian?

thank you in advance