Exploring the Symmetry of Initial Conditions in Progressive Wave Equations

[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.

 

[ShayanJ]

k can be negative!

 

[Nikhil Rajagopalan]

Thank you ShayanJ. How can k - being equal to 2π/λ be a negative value. I am wondering how the single wave equation will explain two different initial waves, pushed into a same medium from the right to the left.

 

[ShayanJ]

Thank you ShayanJ. How can k - being equal to 2π/λ be a negative value. I am wondering how the single wave equation will explain two different initial waves, pushed into a same medium from the right to the left.

You just assume the proper sign for k so that it can give you the desired direction for the motion of the wave!
So actually $k=\pm \frac{2\pi}\lambda$, depending on the direction of motion of the wave.

 

[Nikhil Rajagopalan]

ShayanJ, how could we take the liberty of assuming the sign for k. Even if we take k negative, the equation comes down to Asin(ωt -kx).which is again that for a progressive wave moving towards right. I am sorry if i am missing something very simple.

 

[ShayanJ]

Well, in three dimensions, k becomes a vector, $\vec k$, which is called the wave vector and is responsible for the direction of motion of the plane wave. In one dimension there are only two directions and they're distinguished by sign!

 

[Nikhil Rajagopalan]

I understand that ShayanJ. But what makes me wonder is that i can't find the symmetry here. In case of the wave moving towards right, the argument of the sign function perfectly explains two different initial conditions. But the equation for waves moving towards left looks as though it does not accommodate the possibility that two different waves started by moving the starting end of the rope in two different directions about the mean position has two distinguishable equations as in the case of the one moving towards right.

 

[Nikhil Rajagopalan]

Thank you ShayanJ for the help. I think i may have an explanation now. Considering two initial waves. A sin(kx) and the second one - A sin(kx). Moving towards left, substituting x+vt for x in both, the two distinct equations are A sin (kx +ωt) (for an observer waiting at a point along the line of propagation, mountain/crest hits first) and - A sin (kx +ωt) (for an observer waiting at a point, valley/trough hits first) .
As for the same two initial functions moving towards right, substituting x-vt for x in both, the two distinct equations are A sin (kx -ωt) (for an observer waiting at a point, valley/trough hits first) and - A sin (kx -ωt) - which can be written as A sin (ωt - kx) (for an observer waiting at a point along the line of propagation, crest/mountain hits first).
In a different way of reasoning, with two position for the signs (1.inside the argument, 2.outside the sign function) and each position to be filled with 2 signs, a plus or a minus. There will anyways be 4 possible ways of doing it. Making it 1 in a direction each for 2 initial conditions, symmetrically.

I assume, this was a very elementary thing,i was not getting it right. I appreciate your help.