Trig identity in superposition

[jwxie]

Homework Statement

For two waves in superposition, we have, let say
[tex]\[y_{1} = Asin(kx-wt),
y_{2} = Asin(kx-wt+\Phi )\][/tex]

We use the trig identity sin A + sin B to simplify the combination

Whereas for standing waves, we use the trig identity sin(a +/- b) to combine the result.

As I was going over the note today I noticed that, and this is bothering so I want to know why we use sin (a +/- b) for the standing wave? Can we produce the same result for standing wave y = 2A sin(kx) cos(wt) using the other trig identity I mentioned?

Thank you for any input!

Homework Equations

The Attempt at a Solution

 

[jbsteele]

Yes, you can use the other trigonometric identity to produce the same result for a standing wave. The two identities are related, since sin(A + B) = sin(A)cos(B) + cos(A)sin(B), and sin(A - B) = sin(A)cos(B) - cos(A)sin(B). So, the combination of two waves in superposition can be written as: \[y_{1}+y_{2}=2A\sin(kx-wt)\cos(\Phi)+2A\cos(kx-wt)\sin(\Phi )\]If you set \[\Phi=wt\] then you can rewrite this as: \[y_{1}+y_{2}=2A\sin(kx)\cos(wt)+2A\cos(kx)\sin(wt)\]which is the same as the standing wave equation: \[y=2A\sin(kx)\cos(wt)\]