Wave packet

here is wave equation:

I know that any function of x-ct variable will solve the above equation, where c is wave velocity.

Below is an example of wave written as wave packet with Ai - Airy function.

the first part of equation describes propagation of wave' envelope with group velocity and second part, Airy function, describes propagation of wave itself with phase velocity.

I noticed that both parts can't be written as

part1(x-Vg*t) and part2(x-Vph*t) .... so are these parts still represent traveling waves?

If so then is it true that any function f(x,t) with x appearing only once in power 1, will describe travelling wave?

thanks