Deriving the progressive mechanical wave equation

[Nikhil Rajagopalan]

Is it correct to state that a progressive wave, originates when a simple harmonic motion is imparted continuously to adjacent particles from one direction to another moving with a velocity v. Using this idea, substituting (t - x/v) instead of t is the simple harmonic motion function y=Asin(ωt), we obtain the final answer as y= Asin(ωt - kx).

In another method, drawing a sine wave and finding out the function for a sine wave propagating towards right with a velocity v , substituting (x - vt) instead of x, the propagating wave function is obtained as y= Asin(kx- ωt).

Why is there a phase difference here?

 

[Simon Bridge]

Notice: the first one gets you $y(x,t)=-A\sin (kx-\omega t)$
If you want the same phase, then the first sub should be $x/v - t \to t$, or use a cosine wave.

The difference is because of how the wave propagates.
Think of the physical situation being described in each case: In the first you grab a point, say at x=0, and wave it first up and then down (and suppress the -x propagating solution); in the second you have a wave already and you shove it to one side - so that x=0 goes down first and then up.
The maths is just describing that correctly.

 

[Nikhil Rajagopalan]

Thank you Simon Bridge, the answer was really helpful.

 

[Nikhil Rajagopalan]

For the wave traveling towards left, the equations is Asin(kx + ωt). How does the same mathematical equation explain the possibility of two initial conditions. In the case of the wave traveling towards right, Asin(kx - ωt) and Asin(ωt - kx) gives two initial conditions Asin(kx) and -Asin(kx) on substituting t=0. Explaining the possibility of two different initial movements. In the case of waves traveling towards left, this difference doesn't come up. How can we explain this situation physically.