The wave equation for a disturbance $y(x,t)$ is:
$$ \frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2} $$
Consider the propagation of a gaussian pulse on a string
$$ y(x) = A \exp \left ( -\frac{x^2}{a^2} \right ) $$
Where $a$ takes a constant value
We can move the pulse by $\pm$ b by doing the following
$$ +b \rightarrow y(x) = A \exp \left ( -\frac{(x-b)^2}{a^2} \right ) \qquad ;\qquad -b \rightarrow y(x) = A \exp \left ( -\frac{(x+b)^2}{a^2} \right ) $$
Thus we can define a constant velocity $v$ such that
$$ y(x,t) = A \exp \left ( -\frac{(x-vt)^2}{a^2} \right ) $$
$$ y(x,t) = A \sin \left [ \frac{2 \pi}{\lambda} (x-vt) \right ] $$
https://www.desmos.com/calculator/vro1yse8pp
$y(x,0)=A\sin \left ( \frac{2\pi x}{\lambda} \right )$
angular frequency $\omega = 2 \pi \nu$
velocity of propagation $v=\frac{\omega}{k}$
frequency $\nu = \frac{v}{\lambda}$
wavenumber $k=\frac{2\pi}{\lambda}$
$\boxed{y(x,t)=A\sin(kx-\omega t)}$
$$ y(x,t) = f(x-vt) + g(x+vt) $$