What is the difference between the old and new wave function equations?

[QuarkCharmer]

Homework Statement

Not a homework problem, just a general question that I find confusing.

Homework Equations

The Attempt at a Solution

So, back in Trigonometry and Classical Mechanics I learned that the equation that best represents a wave. Now, Solving the differential equation that is the wave function gives this:
$$f(x,t)=Y_{m}cos(kx-ωt)$$

What has me confused is the new "phase", specifically that it's not ωt+$\phi$.

I gather that k is the wave number, well, the number of wavelengths per unit distance, x is the distance down the wave of course. ω is now the Angular Frequency (not velocity!), which is simply the number of rotations per unit time more or less. So if ω is 2π over the period, what exactly is that other t representing? Seems like it's unit would cancel with ω which is clearly desired, but I just don't see how this whole thing is working exactly.

I can do any of the rote number-plugging homework problems assigned to me, but I simply don't see the relationship between this newer (for me) wave funtion and the one with $\phi$ that I am more familiar with. I can't even find a resource outside my text pertaining to this equation, all searches yield the equation with phi.

 

[Spinnor]

Phi just shifts time or position a bit, if you set your clocks right you can make phi go away.

Your old function which just involved time might describe the position of a vibrating mass connected to a spring.

The new function might describe a wave on a vibrating string, the height of the string now depends on both time and where you are along the string.