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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.

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.

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.