Question

The equation of wave moving in negative x-direction

Answer 1

y = A Cos (ω t - k x + Ф)   is the equation of a wave that is travelling in positive x direction with a velocity  v.
             ω = frequency and is positive.
                 k = wave number and is positive.
             Ф = initial phase value at t = 0 at origin x = 0.

So  at t = 0, at x = X1,       y = A Cos (Ф -  k X1)

At t = Δt , at X1 +Δx,  let  y be the same, or the phase of the wave be the same:

  hence,   y = A Cos (ω Δt - k X1 - k Δx + Ф) =  A Cos (Ф - k X1)
                 so  ω Δt =  k Δx
                    Δx / Δt = ω / k  = velocity of the wave in the + ve x direction.

  As the point that has the same constant phase value, travels in +ve x direction with the above speed.

Then for getting the velocity of the wave in the negative x direction, replace x by -x.

     y = A Cos (ω t + k x +Ф)

Find the value of  y  at t= 0 and at a particular x = X1.
Then  find the value of      Δx  such that at t = Δt, and X1 + Δx the phase value of the wave is the same.  Then  find Δx/Δt.  If it is negative, then the wave is travelling in the negative x direction.