Waves on a string

[aks_sky]

1. The problem statement, all variables and given/known data
The displacement y of a long string at position x and time t is given by

y(x,t) = cos (kx-wt) + sin (kx-wt)

Show that the motion of the string at any point is SHM.


3. The attempt at a solution

As far as i know this is something to do with adding 2 waves together and in this case we have a sinusoidal wave and a cosine wave. If one wave is the reflection of the other then a standing wave will develop, and i think that in this case a standing wave will develop and that would mean that the motion of the string is SHM.

Is that correct? Would there be a better explanation as to why the motion is in SHM.

p.s not homework, just past exam question.

[hikaru1221]

To prove the motion a SHM, just transform the displacement into the form of cosine or sine. In this problem, I think it's completely about mathematics.
How would you prove that this is standing wave by the way?

[Redbelly98]

3. The attempt at a solution

As far as i know this is something to do with adding 2 waves together and in this case we have a sinusoidal wave and a cosine wave. If one wave is the reflection of the other then a standing wave will develop, and i think that in this case a standing wave will develop and that would mean that the motion of the string is SHM.

Is that correct? Would there be a better explanation as to why the motion is in SHM.

p.s not homework, just past exam question.
For a standing wave, you must have two waves traveling in opposite directions -- not the case here, since both waves travel in the +x direction.

You could try the angle-addition formulas for sine and cosine, and work through the algebra.

Hope that helps.