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).
I don't see the logic and I never seen any book that presented it this way. I disagree with the book. In fact, I am at this very moment doing a lot of digging and asking questions regarding to these very kind of phasing issue with respect to direction of propagation, been searching through a lot of books and no body tries to put the equation like this way.......as if it is not confused enough dealing with phase constant with respects to t and x alone.
they just come from simple manipulation,there is nothing special about it.Don't break your head on this.