What is the difference between using $y=A\sin\omega t$ and $y=A\sin(ct-x)$ in a wave formula?
I am not a math student and I am not getting this.
When you use the equation $y(t)=A \sin(\omega t)$ to describe the motion of a particle what you get is the oscillation of a particle located at a particular point in space and just vibrating in the perpendicular direction to and fro. But when you use $y(x,t)=A \sin(\omega t-kx)$ you assign time-varying $y$ coordinate values to each particle in space . As at any point say $x=x^0$ we can substitute it in equation and obtain the equation of motion of a particle if it was located at $x=x^0$. To summarise I just want to say that the first equation is just a dime dependent equation of motion but the second equation is varying with both time and $x$ coordinate and if you observe a particular point only then you fix the value of $x$ and both equations can be used after phase correction