y(x,t)=Asin(ωt+kx) is the equation of motion for a simple harmonic oscillator.
You get this by solving Newton's force law..
[itex] F=ma=kx \\
ma+kx=0 \\
a+\frac{k}{m}x=0 [/itex]
Or you can write
[itex] \ddot{x}+\frac{k}{m}x=0 [/itex]
This is a differential equation, solved by Asin(ωt+kx), where [itex] \omega = \sqrt{\frac{k}{m}} [/itex].
I'm not sure if this answers your question?
