Why does the standing wave equation depend on which waves you add?

[j-e_c]

My lecturer said that a standing wave is formed when two waves that travel in the opposite have the same frequency.

He said that if the waves are y₁ and y₂, then the resulting wave y can be given as the sum:

y = y₁ + y₂.

y = Asin($$\omega$$t - kx) + Asin($$\omega$$t + kx). (1)

Where the plus and minus kx denotes their direction.

However, when (with a bit of trigonometric identity work) equation (1) is simplified it gives:

y = 2Asin($$\omega$$t)cos(kx).

But how can this be? I mean, if x = 0, then the equation tells us that there is an antinode, which (for a string) isn't true.

I've seen the equation y = 2Acos($$\omega$$t)sin(kx), which makes more sense when I consider a string for example.

To get to it you need the equation

y = Asin(kx - $$\omega$$t) + Asin(kx + $$\omega$$t) (2)

My question is, why would you use equation (2) and not equation (1)?

Thank you in advance for your help!

 

[j-e_c]

It's OK, I've solved it now.