When Do We Use Each SHM Equation?

[negation]

Let's say there are 2 equations: (1) y(x,t) = A sin (ωt - ø) and (2) y2(x,t) = A sin (kx - vt)

When are we interested in one over the other? Obviously, (1) tells us that y is represented in terms of ωt( 2πt/T) and ø. Whereas, (2) produces an equation stating the wave number, k(I tend to look at k as the number of cycles/2π) and if we take 2π/k, we obtain the period, T.
vt gives us the shift/ distance as a function of the speed of the wave and the time over which it travels. vt is also really just ø, isn't it?

 

[dauto]

The second equation is actually a wave equation while the first is just a SHM equation - there is no x on the right hand side of the first equation. The 2nd equation should actually be written 2(x,t) = A sin (kx - ωt). 2π/k is actually the wavelength, not the period. ø is just an arbitrary constant phase. ωt cannot be identified with a constant phase since it is a function of time.