# I Deriving the progressive mechanical wave equation

1. Jan 12, 2017

### 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?

2. Jan 12, 2017

### 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.

3. Jan 12, 2017