Quote by Sciencer
we have the wave equation as follows with non zero phase constant:
y(x,t) = ym * sin(k( x  PHI/k)  wt)
or
y(x,t) = ym * sin(kx  w(t + PHI / w))
I don't understand where did the PHI /k or PHI / w came from ??
I understand how did we derive the wave equation but I don't understand this part.

You just substitute in and both equation are the same.
But the more basic thing is, I never seen any book write it this way, that is very confusing. The three terms are totally independent. [itex]\omega t[/itex] is the time dependent, kx is distance dependent, and [itex] \phi[/itex] is a phase constant. You don't confuse this more by mixing them together as if they are related.
People usually set either t=0 or x=0 as a reference and generate two separate equations that relate t or x with [itex]\phi[/itex]. With this, you can generate two separate graphs of (y vs t) or (y vs x).