# Constructive interference derivation

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.

• I suggest you to take the function $y(x,t)=A\sin(ct-x)$ and plot it for different values of $t$. You'll then see what the $x$ dependence does. – eranreches Nov 28 '18 at 17:54
• But does it make a huge difference if I use wt in my derivation instead of ct-x? – Faaiza Ibrahim Nov 28 '18 at 17:56
• These are different things. Note that the first is independent of the position coordinate - so if you're trying to calculate the interference pattern at a specific point in space, then that's what you need. – eranreches Nov 28 '18 at 17:57
• The first is constant in space, the second is not. Am I understanding this correctly? – Aaron Stevens Nov 28 '18 at 18:01

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